{ "cells": [ { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\left[x = -5 \\, y + 9\\right]\\)" ], "text/latex": [ "$\\displaystyle \\left[x = -5 \\, y + 9\\right]$" ], "text/plain": [ "[x == -5*y + 9]" ] }, "execution_count": 4, "metadata": { }, "output_type": "execute_result" } ], "source": [ ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "

Analiza matematyczna - PS

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Informatyka, sem.I, studia niestacjonarne I stopnia, 2023/24

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Lista nr 1: Funkcje jednej zmiennej. Własności funkcji

" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Zad. 1. Sprawdzić, czy podane funkcje są rosnące na wskazanych zbiorach:\n", "\n", "a) $f(x)=x^2,\\ x\\in \\langle 0;\\infty)$

\n", "b) $g(x)=\\displaystyle\\frac{1}{x^4+1},\\ x\\in(-\\infty;0\\rangle$

\n", "c) $h(x)=\\sqrt[3]{x},\\ x\\in(-\\infty;0\\rangle;$

\n", "d) $p(x)=\\sqrt{x+1},\\ x\\in\\langle -1;\\infty)$

\n", "\n" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "Graphics object consisting of 1 graphics primitive" ] }, "execution_count": 8, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#a\n", "x1,x2 = var('x1,x2')\n", "assume(x1=0)\n", "\n", "f(x) = x^2\n", "bool(f(x1)\\(\\displaystyle \\mathrm{False}\\)" ], "text/latex": [ "$\\displaystyle \\mathrm{False}$" ], "text/plain": [ "False" ] }, "execution_count": 15, "metadata": { }, "output_type": "execute_result" }, { "name": "stdout", "output_type": "stream", "text": [ "verbose 0 (3935: plot.py, generate_plot_points) WARNING: When plotting, failed to evaluate function at 100 points.\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "verbose 0 (3935: plot.py, generate_plot_points) Last error message: 'Unable to compute f(-0.001334360219118633)'\n" ] }, { "data": { "image/png": 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", "text/plain": [ "Graphics object consisting of 1 graphics primitive" ] }, "execution_count": 15, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#c\n", "forget()\n", "x1,x2 = var('x1,x2')\n", "assume(x1=-1)\n", "\n", "f(x) = (x+1)**(1/2)\n", "bool(f(x1)-f(x2)<0)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Zad. 2. Sprawdzić, czy podane funkcje są malejące na wskazanych zbiorach:\n", "\n", "a) $f(x)=3-4x,\\ x\\in\\mathbb{R}$

\n", "b) $g(x)=x^2-2x,\\ x\\in(-\\infty;1\\rangle$

\n", "c) $h(x)=\\displaystyle\\frac{1}{1+x^2},\\ x\\in\\langle 0;\\infty)$

\n", "d) $p(x)=\\displaystyle\\frac{1}{1+x},\\ x\\in(-\\infty;-1)$

\n", "\n" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "True" ] }, "execution_count": 17, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#a\n", "forget()\n", "x1,x2 = var('x1,x2')\n", "assume(x1f(x2))" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "True" ] }, "execution_count": 22, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#b\n", "reset()\n", "forget()\n", "x1,x2 = var('x1,x2')\n", "assume(x1g(x2))\n" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "True" ] }, "execution_count": 14, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#c\n", "forget()\n", "x1,x2 = var('x1,x2')\n", "assume(x10)\n", "\n", "h(x) = 1/(1+x^2)\n", "bool(h(x1)-h(x2)>0)" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "True" ] }, "execution_count": 13, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#d\n", "forget()\n", "x1,x2 = var('x1,x2')\n", "assume(x1-1)\n", "\n", "p(x) = 1/(1+x)\n", "bool(p(x1)-p(x2)>0)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "\n", "Zad. 3. Sprawdzić, czy podane funkcje są różnowartościowe na wskazanych\n", "zbiorach:\n", "\n", "a) $f(x)=x^3+1,\\ x\\in\\mathbb{R}$

\n", "b) $g(x)=\\displaystyle\\frac{1}{x^2},\\ x\\in(-\\infty;0)$

\n", "c) $h(x)=\\sqrt{x}+1,\\ x\\in\\langle 0;\\infty)$

" ] }, { "cell_type": "code", "execution_count": 33, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\mathrm{False}\\)" ], "text/latex": [ "$\\displaystyle \\mathrm{False}$" ], "text/plain": [ "False" ] }, "execution_count": 33, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "\\(\\displaystyle \\left[x_{1} = \\frac{1}{2} \\, x_{2} {\\left(i \\, \\sqrt{3} - 1\\right)}, x_{1} = -\\frac{1}{2} \\, x_{2} {\\left(i \\, \\sqrt{3} + 1\\right)}, x_{1} = x_{2}\\right]\\)" ], "text/latex": [ "$\\displaystyle \\left[x_{1} = \\frac{1}{2} \\, x_{2} {\\left(i \\, \\sqrt{3} - 1\\right)}, x_{1} = -\\frac{1}{2} \\, x_{2} {\\left(i \\, \\sqrt{3} + 1\\right)}, x_{1} = x_{2}\\right]$" ], "text/plain": [ "[x1 == 1/2*x2*(I*sqrt(3) - 1), x1 == -1/2*x2*(I*sqrt(3) + 1), x1 == x2]" ] }, "execution_count": 33, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#a\n", "forget()\n", "x1,x2 = var('x1,x2')\n", "#assume(f(x1)==f(x2))\n", "f(x) = x^3 + 1\n", "# równoważnoc implikacji a=>b <=> ~a=>~b\n", "show(bool(x1==x2))\n", "\n", "#assume(x1==x2)\n", "#show(bool(f(x1)==f(x2)))\n", "\n", "show(solve(f(x1)==f(x2),x1))" ] }, { "cell_type": "code", "execution_count": 34, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "False" ] }, "execution_count": 34, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#b\n", "forget()\n", "x1,x2 = var('x1,x2')\n", "assume(x1!=x2, x1<0)\n", "\n", "g(x) = 1/(x^2)\n", "bool(g(x1)!=g(x2))" ] }, { "cell_type": "code", "execution_count": 35, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "True" ] }, "execution_count": 35, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#c\n", "forget()\n", "x1,x2 = var('x1,x2')\n", "assume(x1!=x2, x1>=0)\n", "\n", "g(x) = x**1/2 + 1\n", "bool(g(x1)!=g(x2))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Zad. 4. Sprawdzić, które z podanych funkcji są parzyste, a które nieparzyste:\n", "\n", "a) $\\displaystyle f(x)=x^4-3x^2+1$

\n", "b) $\\displaystyle g(x)=2^x+2^{-x}$

\n", "c) $\\displaystyle h(x)=|\\sin x|$

\n", "d) $\\displaystyle p(x)=\\frac{\\sin x}{x^3}$

\n", "e) $\\displaystyle f(x)=\\frac{2+x^2}{x^5},$

\n", "f) $\\displaystyle g(x)=\\sin^3 x$

\n", "g) $\\displaystyle h(x)=3^x-3^{-x}$

\n", "h) $\\displaystyle p(x)=x|x|$

" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "True" ] }, "execution_count": 4, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#a\n", "forget()\n", "x1,x2 = var('x1,x2')\n", "assume(x1==-x2)\n", "\n", "g(x) = x^4 -3*x^2 + 1\n", "bool(g(x1)==g(x2))" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "False\n", "False\n" ] } ], "source": [ "#b\n", "forget()\n", "x1,x2 = var('x1,x2')\n", "assume(x1==-x2)\n", "\n", "g(x) = 2^x -2^-x\n", "print(bool(g(x1)==g(x2)))\n", "print(bool(g(x1)==-g(-x2)))" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "False" ] }, "execution_count": 8, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#c\n", "forget()\n", "x1,x2 = var('x1,x2')\n", "assume(x1==-x2)\n", "\n", "g(x) = abs(sin(x))\n", "bool(g(x1)==-g(-x2))" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "False" ] }, "execution_count": 2, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#d\n", "forget()\n", "x1,x2 = var('x1,x2')\n", "assume(x1==-x2)\n", "\n", "g(x) = sin(x)/(x^3)\n", "bool(g(x1)==-g(-x2))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Zad. 5. Określić funkcje złożone $f\\circ f$, $f\\circ g$, $g\\circ f$,\n", "$g\\circ g$ oraz ich dziedziny, jeżeli:\n", "\n", "a) $\\displaystyle f(x)=|x|, \\ g(x)=-3x+2$

\n", "b) $\\displaystyle f(x)=\\sqrt{x},\\ g(x)=x^3+1$

\n", "c) $\\displaystyle f(x)=x^2,\\ g(x)=\\sqrt{x},$

\n", "d) $\\displaystyle f(x)=2^x,\\ g(x)=\\cos x$

\n", "e) $\\displaystyle f(x)=x^3,\\ g(x)=\\frac{1}{\\sqrt[3]{x}}$

\n", "f) $\\displaystyle f(x)=\\frac{x}{1+x^2},\\ g(x)=\\frac{1}{x},$

\n", "g) $\\displaystyle f(x)=\\log x,\\ g(x)=x^2+1.$\n", "\n" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle {\\left| x \\right|}\\)" ], "text/latex": [ "$\\displaystyle {\\left| x \\right|}$" ], "text/plain": [ "abs(x)" ] }, "execution_count": 14, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "\\(\\displaystyle {\\left| -3 \\, x + 2 \\right|}\\)" ], "text/latex": [ "$\\displaystyle {\\left| -3 \\, x + 2 \\right|}$" ], "text/plain": [ "abs(-3*x + 2)" ] }, "execution_count": 14, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "\\(\\displaystyle -3 \\, {\\left| x \\right|} + 2\\)" ], "text/latex": [ "$\\displaystyle -3 \\, {\\left| x \\right|} + 2$" ], "text/plain": [ "-3*abs(x) + 2" ] }, "execution_count": 14, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "\\(\\displaystyle 9 \\, x - 4\\)" ], "text/latex": [ "$\\displaystyle 9 \\, x - 4$" ], "text/plain": [ "9*x - 4" ] }, "execution_count": 14, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#a\n", "forget()\n", "x1,x2 = var('x1,x2')\n", "assume(x1==-x2)\n", "\n", "f(x) = abs(x)\n", "g(x) = -3*x+2\n", "\n", "ff(x) = f(f(x))\n", "show(ff(x))\n", "#x nalezy do R\n", "#y nalezy do +R\n", "fg(x) = f(g(x))\n", "show(fg(x))\n", "#x nalezy do R\n", "#y nalezy do R\n", "gf(x) = g(f(x))\n", "show(gf(x))\n", "#x nalezy do R\n", "#y nalezy do R>-1\n", "gg(x) = g(g(x))\n", "show(gg(x))" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle x^{\\frac{1}{4}}\\)" ], "text/latex": [ "$\\displaystyle x^{\\frac{1}{4}}$" ], "text/plain": [ "x^(1/4)" ] }, "execution_count": 8, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "\\(\\displaystyle \\sqrt{x^{3} + 1}\\)" ], "text/latex": [ "$\\displaystyle \\sqrt{x^{3} + 1}$" ], "text/plain": [ "sqrt(x^3 + 1)" ] }, "execution_count": 8, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "\\(\\displaystyle x^{\\frac{3}{2}} + 1\\)" ], "text/latex": [ "$\\displaystyle x^{\\frac{3}{2}} + 1$" ], "text/plain": [ "x^(3/2) + 1" ] }, "execution_count": 8, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "\\(\\displaystyle {\\left(x^{3} + 1\\right)}^{3} + 1\\)" ], "text/latex": [ "$\\displaystyle {\\left(x^{3} + 1\\right)}^{3} + 1$" ], "text/plain": [ "(x^3 + 1)^3 + 1" ] }, "execution_count": 8, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#b\n", "forget()\n", "x1,x2 = var('x1,x2')\n", "assume(x1==-x2)\n", "\n", "f(x) = sqrt(x)\n", "g(x) = x^3 + 1\n", "\n", "ff(x) = f(f(x))\n", "show(ff(x))\n", "\n", "fg(x) = f(g(x))\n", "show(fg(x))\n", "\n", "gf(x) = g(f(x))\n", "show(gf(x))\n", "\n", "gg(x) = g(g(x))\n", "show(gg(x))" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle x^{4}\\)" ], "text/latex": [ "$\\displaystyle x^{4}$" ], "text/plain": [ "x^4" ] }, "execution_count": 13, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "\\(\\displaystyle x\\)" ], "text/latex": [ "$\\displaystyle x$" ], "text/plain": [ "x" ] }, "execution_count": 13, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "\\(\\displaystyle \\sqrt{x^{2}}\\)" ], "text/latex": [ "$\\displaystyle \\sqrt{x^{2}}$" ], "text/plain": [ "sqrt(x^2)" ] }, "execution_count": 13, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "\\(\\displaystyle x^{\\frac{1}{4}}\\)" ], "text/latex": [ "$\\displaystyle x^{\\frac{1}{4}}$" ], "text/plain": [ "x^(1/4)" ] }, "execution_count": 13, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#c\n", "forget()\n", "x1,x2 = var('x1,x2')\n", "assume(x1==-x2)\n", "\n", "f(x) = x^2\n", "g(x) = sqrt(x)\n", "\n", "ff(x) = f(f(x))\n", "show(ff(x))\n", "\n", "fg(x) = f(g(x))\n", "show(fg(x))\n", "\n", "gf(x) = g(f(x))\n", "show(gf(x))\n", "\n", "gg(x) = g(g(x))\n", "show(gg(x))" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle 2^{\\left(2^{x}\\right)}\\)" ], "text/latex": [ "$\\displaystyle 2^{\\left(2^{x}\\right)}$" ], "text/plain": [ "2^(2^x)" ] }, "execution_count": 11, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "\\(\\displaystyle 2^{\\cos\\left(x\\right)}\\)" ], "text/latex": [ "$\\displaystyle 2^{\\cos\\left(x\\right)}$" ], "text/plain": [ "2^cos(x)" ] }, "execution_count": 11, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "\\(\\displaystyle \\cos\\left(2^{x}\\right)\\)" ], "text/latex": [ "$\\displaystyle \\cos\\left(2^{x}\\right)$" ], "text/plain": [ "cos(2^x)" ] }, "execution_count": 11, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "\\(\\displaystyle \\cos\\left(\\cos\\left(x\\right)\\right)\\)" ], "text/latex": [ "$\\displaystyle \\cos\\left(\\cos\\left(x\\right)\\right)$" ], "text/plain": [ "cos(cos(x))" ] }, "execution_count": 11, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#d\n", "forget()\n", "x1,x2 = var('x1,x2')\n", "assume(x1==-x2)\n", "\n", "f(x) = 2**x\n", "g(x) = cos(x)\n", "\n", "ff(x) = f(f(x))\n", "show(ff(x))\n", "\n", "fg(x) = f(g(x))\n", "show(fg(x))\n", "\n", "gf(x) = g(f(x))\n", "show(gf(x))\n", "\n", "gg(x) = g(g(x))\n", "show(gg(x))" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle x^{9}\\)" ], "text/latex": [ "$\\displaystyle x^{9}$" ], "text/plain": [ "x^9" ] }, "execution_count": 10, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "\\(\\displaystyle \\frac{1}{x}\\)" ], "text/latex": [ "$\\displaystyle \\frac{1}{x}$" ], "text/plain": [ "1/x" ] }, "execution_count": 10, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "\\(\\displaystyle \\frac{1}{{\\left(x^{3}\\right)}^{\\frac{1}{3}}}\\)" ], "text/latex": [ "$\\displaystyle \\frac{1}{{\\left(x^{3}\\right)}^{\\frac{1}{3}}}$" ], "text/plain": [ "(x^3)^(-1/3)" ] }, "execution_count": 10, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "\\(\\displaystyle x^{\\frac{1}{9}}\\)" ], "text/latex": [ "$\\displaystyle x^{\\frac{1}{9}}$" ], "text/plain": [ "x^(1/9)" ] }, "execution_count": 10, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#e ???? potegasx\n", "forget()\n", "x1,x2 = var('x1,x2')\n", "assume(x1==-x2)\n", "\n", "f(x) = x^3\n", "g(x) = 1/(x**(1/3))\n", "\n", "ff(x) = f(f(x))\n", "show(ff(x))\n", "\n", "fg(x) = f(g(x))\n", "show(fg(x))\n", "\n", "gf(x) = g(f(x))\n", "show(gf(x))\n", "\n", "gg(x) = g(g(x))\n", "show(gg(x))" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\frac{x}{{\\left(x^{2} + 1\\right)} {\\left(\\frac{x^{2}}{{\\left(x^{2} + 1\\right)}^{2}} + 1\\right)}}\\)" ], "text/latex": [ "$\\displaystyle \\frac{x}{{\\left(x^{2} + 1\\right)} {\\left(\\frac{x^{2}}{{\\left(x^{2} + 1\\right)}^{2}} + 1\\right)}}$" ], "text/plain": [ "x/((x^2 + 1)*(x^2/(x^2 + 1)^2 + 1))" ] }, "execution_count": 3, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "\\(\\displaystyle \\frac{1}{x {\\left(\\frac{1}{x^{2}} + 1\\right)}}\\)" ], "text/latex": [ "$\\displaystyle \\frac{1}{x {\\left(\\frac{1}{x^{2}} + 1\\right)}}$" ], "text/plain": [ "1/(x*(1/x^2 + 1))" ] }, "execution_count": 3, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "\\(\\displaystyle \\frac{x^{2} + 1}{x}\\)" ], "text/latex": [ "$\\displaystyle \\frac{x^{2} + 1}{x}$" ], "text/plain": [ "(x^2 + 1)/x" ] }, "execution_count": 3, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "\\(\\displaystyle x\\)" ], "text/latex": [ "$\\displaystyle x$" ], "text/plain": [ "x" ] }, "execution_count": 3, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#f\n", "forget()\n", "x = var('x')\n", "\n", "f(x) = x/(1+x^2)\n", "g(x) = 1/x\n", "\n", "ff(x) = f(f(x))\n", "show(ff(x))\n", "\n", "fg(x) = f(g(x))\n", "show(fg(x))\n", "\n", "gf(x) = g(f(x))\n", "show(gf(x))\n", "\n", "gg(x) = g(g(x))\n", "show(gg(x))" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\log\\left(\\log\\left(x\\right)\\right)\\)" ], "text/latex": [ "$\\displaystyle \\log\\left(\\log\\left(x\\right)\\right)$" ], "text/plain": [ "log(log(x))" ] }, "execution_count": 15, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "\\(\\displaystyle \\log\\left(x^{2} + 1\\right)\\)" ], "text/latex": [ "$\\displaystyle \\log\\left(x^{2} + 1\\right)$" ], "text/plain": [ "log(x^2 + 1)" ] }, "execution_count": 15, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "\\(\\displaystyle \\log\\left(x\\right)^{2} + 1\\)" ], "text/latex": [ "$\\displaystyle \\log\\left(x\\right)^{2} + 1$" ], "text/plain": [ "log(x)^2 + 1" ] }, "execution_count": 15, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "\\(\\displaystyle {\\left(x^{2} + 1\\right)}^{2} + 1\\)" ], "text/latex": [ "$\\displaystyle {\\left(x^{2} + 1\\right)}^{2} + 1$" ], "text/plain": [ "(x^2 + 1)^2 + 1" ] }, "execution_count": 15, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#g\n", "forget()\n", "x1,x2 = var('x1,x2')\n", "assume(x1==-x2)\n", "\n", "f(x) = log(x)\n", "g(x) = x^2 + 1\n", "\n", "ff(x) = f(f(x))\n", "show(ff(x))\n", "#x nalezy do +R (0,niesk) i log(x) > 0 wiec x > 1 (1,niesk)\n", "fg(x) = f(g(x))\n", "show(fg(x))\n", "#x nalezy do R\n", "gf(x) = g(f(x))\n", "show(gf(x))\n", "#x nalezy do R (0,niesk)\n", "gg(x) = g(g(x))\n", "show(gg(x))\n", "#x nalezy do R" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Zad. 6. Znaleźć funkcję odwrotną do podanej:\n", "\n", "a) $\\displaystyle f(x)=x^2-2x,\\ x\\in\\langle 1;\\infty)$

\n", "b) $\\displaystyle g(x)=2-\\sqrt[5]{x+1},\\ x\\in\\mathbb{R}$

\n", "c) $\\displaystyle h(x)=x^3|x|,\\ x\\in\\mathbb{R},$

\n", "d) $\\displaystyle p(x)=\\left\\{\n", "\\begin{array}{rrr}\n", "3^x & \\mbox{ dla } &x<0\\\\\n", "5^x & \\mbox{ dla } &x\\geqslant 0\n", "\\end{array}\\right.\n", ",\\ x\\in\\mathbb{R}$

\n", "e) $\\displaystyle f(x)=1-3^{-x}$

\n", "f) $\\displaystyle g(x)=x^5+\\sqrt{3},$

\n", "g) $\\displaystyle h(x)=x^6\\mathrm{sgn\\,}x$

\n", "h) $\\displaystyle q(x)=\\left\\{\n", "\\begin{array}{rrr}\n", "-x^2 & \\mbox{ dla } &x<0\\\\\n", "2+x & \\mbox{ dla } &x\\geqslant 0\n", "\\end{array}\\right.\n", ",\\ x\\in\\mathbb{R}$

\n", "i) $\\displaystyle f(x)=\\frac{x}{1+|x|}.$\n", "\n" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\left[x = \\sqrt{y + 1} + 1\\right]\\)" ], "text/latex": [ "$\\displaystyle \\left[x = \\sqrt{y + 1} + 1\\right]$" ], "text/plain": [ "[x == sqrt(y + 1) + 1]" ] }, "execution_count": 19, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#a \n", "forget()\n", "x,y = var('x,y')\n", "assume(x>1)\n", "\n", "f(x) = x^2 - 2*x\n", "show(solve(y==f(x),x))\n", "#f do -1 (x) = sqrt(x+1)+1 dla x>=-1" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\left[x = -5 \\, y + 9\\right]\\)" ], "text/latex": [ "$\\displaystyle \\left[x = -5 \\, y + 9\\right]$" ], "text/plain": [ "[x == -5*y + 9]" ] }, "execution_count": 11, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#b potega z wykladnikiem ? \n", "forget()\n", "x,y = var('x,y')\n", "assume(x>1)\n", "\n", "f(x) = 2 - (x+1)**1/5\n", "show(solve(y==f(x),x))\n", "#f do -1 (x) = sqrt(x+1)+1 dla x>=-1" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\left[x^{3} = \\frac{y}{{\\left| x \\right|}}\\right]\\)" ], "text/latex": [ "$\\displaystyle \\left[x^{3} = \\frac{y}{{\\left| x \\right|}}\\right]$" ], "text/plain": [ "[x^3 == y/abs(x)]" ] }, "execution_count": 9, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#c\n", "forget()\n", "x,y = var('x,y')\n", "\n", "f(x) = (x^3 * abs(x))\n", "show(solve(y==f(x),x))" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\left[x = \\frac{\\log\\left(y\\right)}{\\log\\left(3\\right)}\\right]\\)" ], "text/latex": [ "$\\displaystyle \\left[x = \\frac{\\log\\left(y\\right)}{\\log\\left(3\\right)}\\right]$" ], "text/plain": [ "[x == log(y)/log(3)]" ] }, "execution_count": 6, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "\\(\\displaystyle \\left[x = \\frac{\\log\\left(y\\right)}{\\log\\left(5\\right)}\\right]\\)" ], "text/latex": [ "$\\displaystyle \\left[x = \\frac{\\log\\left(y\\right)}{\\log\\left(5\\right)}\\right]$" ], "text/plain": [ "[x == log(y)/log(5)]" ] }, "execution_count": 6, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#d\n", "forget()\n", "x = var('x')\n", "\n", "assume(x<0)\n", "p1(x) = 3^x\n", "\n", "show(solve(y==p1(x),x))\n", "forget()\n", "assume(x>0)\n", "p2(x) = 5^x\n", "show(solve(y==p2(x),x))\n", "#y=log trojkowy x, y<1\n", "#analogicznie" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\left[x = \\frac{\\log\\left(-\\frac{1}{y - 1}\\right)}{\\log\\left(3\\right)}\\right]\\)" ], "text/latex": [ "$\\displaystyle \\left[x = \\frac{\\log\\left(-\\frac{1}{y - 1}\\right)}{\\log\\left(3\\right)}\\right]$" ], "text/plain": [ "[x == log(-1/(y - 1))/log(3)]" ] }, "execution_count": 12, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#e\n", "forget()\n", "x,y = var('x,y')\n", "\n", "f(x) = 1 - 3^-x\n", "show(solve(y==f(x),x))" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\left[x = \\frac{1}{4} \\, {\\left(y - \\sqrt{3}\\right)}^{\\frac{1}{5}} {\\left(\\sqrt{5} + i \\, \\sqrt{2 \\, \\sqrt{5} + 10} - 1\\right)}, x = -\\frac{1}{4} \\, {\\left(y - \\sqrt{3}\\right)}^{\\frac{1}{5}} {\\left(\\sqrt{5} - i \\, \\sqrt{-2 \\, \\sqrt{5} + 10} + 1\\right)}, x = -\\frac{1}{4} \\, {\\left(y - \\sqrt{3}\\right)}^{\\frac{1}{5}} {\\left(\\sqrt{5} + i \\, \\sqrt{-2 \\, \\sqrt{5} + 10} + 1\\right)}, x = \\frac{1}{4} \\, {\\left(y - \\sqrt{3}\\right)}^{\\frac{1}{5}} {\\left(\\sqrt{5} - i \\, \\sqrt{2 \\, \\sqrt{5} + 10} - 1\\right)}, x = {\\left(y - \\sqrt{3}\\right)}^{\\frac{1}{5}}\\right]\\)" ], "text/latex": [ "$\\displaystyle \\left[x = \\frac{1}{4} \\, {\\left(y - \\sqrt{3}\\right)}^{\\frac{1}{5}} {\\left(\\sqrt{5} + i \\, \\sqrt{2 \\, \\sqrt{5} + 10} - 1\\right)}, x = -\\frac{1}{4} \\, {\\left(y - \\sqrt{3}\\right)}^{\\frac{1}{5}} {\\left(\\sqrt{5} - i \\, \\sqrt{-2 \\, \\sqrt{5} + 10} + 1\\right)}, x = -\\frac{1}{4} \\, {\\left(y - \\sqrt{3}\\right)}^{\\frac{1}{5}} {\\left(\\sqrt{5} + i \\, \\sqrt{-2 \\, \\sqrt{5} + 10} + 1\\right)}, x = \\frac{1}{4} \\, {\\left(y - \\sqrt{3}\\right)}^{\\frac{1}{5}} {\\left(\\sqrt{5} - i \\, \\sqrt{2 \\, \\sqrt{5} + 10} - 1\\right)}, x = {\\left(y - \\sqrt{3}\\right)}^{\\frac{1}{5}}\\right]$" ], "text/plain": [ "[x == 1/4*(y - sqrt(3))^(1/5)*(sqrt(5) + I*sqrt(2*sqrt(5) + 10) - 1), x == -1/4*(y - sqrt(3))^(1/5)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1), x == -1/4*(y - sqrt(3))^(1/5)*(sqrt(5) + I*sqrt(-2*sqrt(5) + 10) + 1), x == 1/4*(y - sqrt(3))^(1/5)*(sqrt(5) - I*sqrt(2*sqrt(5) + 10) - 1), x == (y - sqrt(3))^(1/5)]" ] }, "execution_count": 13, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#f\n", "forget()\n", "x,y = var('x,y')\n", "\n", "f(x) = x^5 + sqrt(3)\n", "show(solve(y==f(x),x))" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\left[x = \\frac{1}{2} \\, {\\left(y - \\mathrm{sgn}\\left(x\\right)\\right)}^{\\frac{1}{6}} {\\left(i \\, \\sqrt{3} + 1\\right)}, x = \\frac{1}{2} \\, {\\left(y - \\mathrm{sgn}\\left(x\\right)\\right)}^{\\frac{1}{6}} {\\left(i \\, \\sqrt{3} - 1\\right)}, x = -{\\left(y - \\mathrm{sgn}\\left(x\\right)\\right)}^{\\frac{1}{6}}, x = -\\frac{1}{2} \\, {\\left(y - \\mathrm{sgn}\\left(x\\right)\\right)}^{\\frac{1}{6}} {\\left(i \\, \\sqrt{3} + 1\\right)}, x = -\\frac{1}{2} \\, {\\left(y - \\mathrm{sgn}\\left(x\\right)\\right)}^{\\frac{1}{6}} {\\left(i \\, \\sqrt{3} - 1\\right)}, x = {\\left(y - \\mathrm{sgn}\\left(x\\right)\\right)}^{\\frac{1}{6}}\\right]\\)" ], "text/latex": [ "$\\displaystyle \\left[x = \\frac{1}{2} \\, {\\left(y - \\mathrm{sgn}\\left(x\\right)\\right)}^{\\frac{1}{6}} {\\left(i \\, \\sqrt{3} + 1\\right)}, x = \\frac{1}{2} \\, {\\left(y - \\mathrm{sgn}\\left(x\\right)\\right)}^{\\frac{1}{6}} {\\left(i \\, \\sqrt{3} - 1\\right)}, x = -{\\left(y - \\mathrm{sgn}\\left(x\\right)\\right)}^{\\frac{1}{6}}, x = -\\frac{1}{2} \\, {\\left(y - \\mathrm{sgn}\\left(x\\right)\\right)}^{\\frac{1}{6}} {\\left(i \\, \\sqrt{3} + 1\\right)}, x = -\\frac{1}{2} \\, {\\left(y - \\mathrm{sgn}\\left(x\\right)\\right)}^{\\frac{1}{6}} {\\left(i \\, \\sqrt{3} - 1\\right)}, x = {\\left(y - \\mathrm{sgn}\\left(x\\right)\\right)}^{\\frac{1}{6}}\\right]$" ], "text/plain": [ "[x == 1/2*(y - sgn(x))^(1/6)*(I*sqrt(3) + 1), x == 1/2*(y - sgn(x))^(1/6)*(I*sqrt(3) - 1), x == -(y - sgn(x))^(1/6), x == -1/2*(y - sgn(x))^(1/6)*(I*sqrt(3) + 1), x == -1/2*(y - sgn(x))^(1/6)*(I*sqrt(3) - 1), x == (y - sgn(x))^(1/6)]" ] }, "execution_count": 14, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#g\n", "forget()\n", "x,y = var('x,y')\n", "\n", "f(x) = x^6 + sign(x)\n", "show(solve(y==f(x),x))" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\left[x = -\\sqrt{y}\\right]\\)" ], "text/latex": [ "$\\displaystyle \\left[x = -\\sqrt{y}\\right]$" ], "text/plain": [ "[x == -sqrt(y)]" ] }, "execution_count": 15, "metadata": { }, "output_type": "execute_result" }, { "data": { "text/html": [ "\\(\\displaystyle \\left[x = y - 2\\right]\\)" ], "text/latex": [ "$\\displaystyle \\left[x = y - 2\\right]$" ], "text/plain": [ "[x == y - 2]" ] }, "execution_count": 15, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#h\n", "forget()\n", "assume(x<0)\n", "p1(x) = (-x)^2\n", "\n", "show(solve(y==p1(x),x))\n", "forget()\n", "assume(x>0)\n", "p2(x) = 2 + x\n", "show(solve(y==p2(x),x))" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\\(\\displaystyle \\left[x = y {\\left| x \\right|} + y\\right]\\)" ], "text/latex": [ "$\\displaystyle \\left[x = y {\\left| x \\right|} + y\\right]$" ], "text/plain": [ "[x == y*abs(x) + y]" ] }, "execution_count": 16, "metadata": { }, "output_type": "execute_result" } ], "source": [ "#i\n", "forget()\n", "x,y = var('x,y')\n", "\n", "f(x) = x/(1+abs(x))\n", "show(solve(y==f(x),x))" ] } ], "metadata": { "kernelspec": { "argv": [ "sage-10.1", "--python", "-m", "sage.repl.ipython_kernel", "--matplotlib=inline", "-f", "{connection_file}" ], "display_name": "SageMath 10.1", "env": { }, "language": "sagemath", "metadata": { "cocalc": { "description": "Open-source mathematical software system", "priority": 10, "url": "https://www.sagemath.org/" } }, "name": "sage-10.1", "resource_dir": "/ext/jupyter/kernels/sage-10.1" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.11.1" } }, "nbformat": 4, "nbformat_minor": 4 }