Chapter 1

sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=2 \sin (x+\pi / 4) $$

Suppose that \(f(x)=x^{2}, x \geq 0\), and \(g(x)=\sqrt{x}, x \geq 0 .\) Typically, \(f \circ g \neq g \circ f\), but this is an example in which the order of composition does not matter. Show that \(f \circ g=g \circ f\).

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line with slope 3 and \(y\) -intercept \((0,2)\)

sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=2 \sin (-x) $$

Suppose that \(f(x)=x^{4}, x \in \mathbf{R} .\) For each of the following functions \(g(x)\), determine whether \((f \circ g)(x)=(g \circ f)(x)\) or not. (a) \(g(x)=x+1, x \in \mathbf{R}\). (b) \(g(x)=\sqrt{x}, x \in \mathbf{R}\). (c) \(g(x)=\frac{1}{x}, x>0\). (d) \(g(x)=-x, x \in \mathbf{R}\). (e) \(g(x)=|x|, x \in \mathbf{R}\).

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line with slope \(-1\) and \(y\) -intercept \((0,5)\)

sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=\cos (\pi x) $$

Use a graphing calculator to graph \(f(x)=x^{2}, x \geq 0\), and \(g(x)=x^{3}, x \geq 0\), together. For which values of \(x\) is \(f(x)>g(x)\) and for which is \(f(x)

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line with slope \(1 / 2\) and \(y\) -intercept \((0,2)\)

sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=-2 \cos (\pi x / 4) $$

Use a graphing calculator to graph \(f(x)=x^{3}, x \geq 0\), and \(g(x)=x^{5}, x \geq 0\), together. When is \(f(x)>g(x)\), and when is \(f(x)

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line with slope \(-1 / 3\) and \(y\) -intercept \((0,1 / 3)\)

Explain how the following functions can be obtained from \(y=x^{2}\) by basic transformations: (a) \(y=x^{2}-2\) (b) \(y=(x-1)^{2}+1\) (c) \(y=-2(x+2)^{2}\)

Graph \(y=x^{n}, x \geq 0\), for \(n=1,2,3\), and 4 in one coordinate system. Where do the curves intersect?

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line with slope \(-2\) and \(x\) -intercept \((1,0)\)

Explain how the following functions can be obtained from \(y=x^{3}\) by basic transformations: (a) \(y=x^{3}-1\) (b) \(y=-x^{3}-1\) (c) \(y=-3(x-1)^{3}\)

(a) Graph \(f(x)=x, x \geq 0\), and \(g(x)=x^{2}, x \geq 0\), together, in one coordinate system. (b) For which values of \(x\) is \(f(x) \geq g(x)\), and for which values of \(x\) is \(f(x) \leq g(x)\) ?

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line with slope 1 and \(x\) -intercept ( 2,0 )

Explain how the following functions can be obtained from \(y=1 / x\) by basic transformations: (a) \(y=1-\frac{1}{x}\) (b) \(y=-\frac{1}{x-1}\) (c) \(y=\frac{x}{x+1}\)

(a) Graph \(f(x)=x^{2}\) and \(g(x)=x^{3}\) for \(x \geq 0\), together, in one coordinate system. (b) Show algebraically that \(x \geq x^{2}\) for \(0 \leq x \leq 1\). (c) Show algebraically that \(x \leq x^{2}\) for \(x \geq 1\).

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line with slope \(-1 / 2\) and \(x\) -intercept \((-1 / 2,0)\)

Explain how the following functions can be obtained from \(y=1 / x^{2}\) by basic transformations: (a) \(y=\frac{1}{x^{2}}+1\) (b) \(y=-\frac{1}{(x+1)^{2}}\) (c) \(y=-\frac{1}{x^{2}}-2\)

Show algebraically that if \(n \geq m, x^{n} \leq x^{m}\), for \(0 \leq x \leq 1\), and \(x^{n} \geq x^{m}\), for \(x \geq 1 .\)

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line with slope \(1 / 5\) and \(x\) -intercept \((-1 / 2,0)\)

Explain how the following functions can be obtained from \(y=e^{x}\) by basic transformations: (a) \(y=e^{x}+3\) (b) \(y=e^{-x}\) (c) \(y=2 e^{x-2}+3\)

(a) Show that \(y=x^{4}, x \in \mathbf{R}\), is an even function. (b) Show that \(y=x^{3}, x \in \mathbf{R}\), is an odd function.

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((2,-3)\) and parallel to \(x+2 y-4=0\)

Explain how the following functions can be obtained from \(y=e^{x}\) by basic transformations: (a) \(y=e^{-x}-1\) (b) \(y=-e^{x}+1\) (c) \(y=-e^{x-3}-2\)

Show that (a) \(y=x^{n}, x \in \mathbf{R}\), is an even function when \(n\) is an even integer. (b) \(y=x^{n}, x \in \mathbf{R}\), is an odd function when \(n\) is an odd integer.

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((1,2)\) and parallel to \(x-2 y+4=0\)

Explain how the following functions can be obtained from \(y=\ln x\) by basic transformations: (a) \(y=\ln (x-1)\) (b) \(y=-\ln x+1\) (c) \(y=\ln (x+3)-1\)

In Example 4 we discussed the GiniSimpson diversity index, which we defined to be the probability that two individuals randomly chosen from a population of two cell types are genetically different. We gave a formula for calculating Gini-Simpson diversity index, \(H\) : $$ H(p)=2 p(1-p) \quad p \in[0,1]. $$ Here \(p\) is the proportion of red individuals in the population. Conversely, we could ask what is the probability that the two individuals are genetically identical. Call this probability \(I(p)\). It is given by $$ I(p)=2 p^{2}-2 p+1. $$ (a) The function \(I(p)\) is known as the Simpson index. Explain why the domain of \(I\) is \(p \in[0,1]\). (b) Use a graphing calculator to plot \(I(p)\). (c) What is the range \(I([0,1])\) ?

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((-1,-1)\) and parallel to the line passing through \((0,2)\) and \((3,0)\)

Explain how the following functions can be obtained fron \(y=\ln x\) by basic transformations: (a) \(y=\ln (1-x)\) (b) \(y=\ln (2+x)-1\) (c) \(y=-\ln (2-x)+1\)

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((2,-1)\) and parallel to the line passing through \((0,-4)\) and \((2,1)\)

Explain how the following functions can be obtained from \(y=\sin x\) by basic transformations: (a) \(\sin (\pi x)\) (b) \(\sin \left(x+\frac{\pi}{4}\right)\) (c) \(-2 \sin (\pi x+1)\)

Suppose that a beetle walks up a tree along a straight line at a constant speed of 1 meter per hour. What distance will the beetle have covered after 1 hour, 2 hours, and 3 hours? Write an equation that expresses the distance (in meters) as a function of the time (in hours), and show that this function is a polynomial of degree 1 .

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((1,4)\) and perpendicular to \(2 y-5 x+7=0\)

Explain how the following functions can be obtained from \(y=\cos x\) by basic transformations: (a) \(y=1+2 \cos x\) (b) \(y=-\cos \left(x+\frac{\pi}{4}\right)\) (c) \(y=-\cos \left(\frac{\pi}{2}-x\right)\)

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((1,-1)\) and perpendicular to \(x-2 y+3=0\)

Find the following numbers on a number line that is on a logarithmic scale (base 10\(): 0.003,0.03,3,5,30,50,1000,3000\), and \(30000 .\)

In Problems 33-36, for each function, find the largest possible domain and determine the range. $$ \text { 33. } f(x)=\frac{1}{1-x} $$

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((5,-1)\) and perpendicular to the line passing through \((-2,1)\) and \((1,-2)\)

Find the following numbers on a number line that is on a logarithmic scale (base 10): \(0.03,0.7,1,2,5,10,17,100,150\), and \(2000 .\)

For each function, find the largest possible domain and determine the range. $$ f(x)=\frac{2 x+1}{(x-2)(x+3)} $$

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((4,-1)\) and perpendicular to the line passing through \((-2,0)\) and \((1,1)\)

Find the following numbers on a number line that is on a logarithmic scale (base 10\(): 10^{2}, 10^{-3}, 10^{-4}, 10^{-7}\), and \(10^{-10}\). (b) Can you find 0 on a number line that is on a logarithmic scale? (c) Can you find negative numbers on a number line that is on a logarithmic scale?

For each function, find the largest possible domain and determine the range. $$ \text { 35. } f(x)=\frac{x-2}{x^{2}-9} $$

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((1,3)\) and parallel to the horizontal line passing through \((3,-1)\)

Find the following numbers on a number line that is on a logarithmic scale (base 10\()\) : (i) \(10^{-3}, 2 \times 10^{-3}, 3 \times 10^{-3}\) (ii) \(10^{-1}, 2 \times 10^{-1}, 3 \times 10^{-1}\) (iii) \(10^{2}, 2 \times 10^{2}, 3 \times 10^{2}\) (b) From your answers to (a), how many units (on a logarithmic scale) is (i) \(2 \times 10^{-3}\) from \(10^{-3}\) (ii) \(2 \times 10^{-1}\) from \(10^{-1}\) and (iii) \(2 \times 10^{2}\) from \(10^{2}\) ? (c) From your answers to (a), how many units (on a logarithmic scale) is (i) \(3 \times 10^{-3}\) from \(10^{-3}\) (ii) \(3 \times 10^{-1}\) from \(10^{-1}\) and (iii) \(3 \times 10^{2}\) from \(10^{2}\) ?