Chapter 3

To nd \(f(g(x))\), apply \(g\) to \(x\) and then use the output of \(g\) as the input of \(f .\) Work from the inside out. Let \(f(x)=x^{2}, g(x)=1 / x\), and \(h(x)=3 x+1\) Worked example: $$f(g(h(x)))=f(g(3 x+1))=f\left(\frac{1}{3 x+1}\right)=\left(\frac{1}{3 x+1}\right)^{2}$$ Find the following. (a) \(f(g(x))\) (b) \(g(f(x))\) (c) \(x h(f(x))\) (d) \(f(h(g(x)))\) (e) \(g(g(w))\) (f) \(h(h(t))\) (g) \(g(f(1 / x))\) (h) \(g(2 h(x-1))\) (i) Show that \(g(g(x)) \neq[g(x)]^{2}\) (j) Show that \([h(x)]^{2} \neq h\left(x^{2}\right)\).

The zeros of the function \(f(x)\) are at \(x=-4,-1,2\), and 8 . What are the zeros of (a) \(m(x)=5 f(x)\) ? (b) \(g(x)=f(x+2)\) ? (c) \(h(x)=f(2 x)\) ? (d) \(j(x)=f(x-1)\) ? Verify your answers analytically.

Let \(f(x)=x^{2}\) and \(g(x)=1 / x\). Use your knowledge of the graphs of \(f\) and \(g\) to sketch the graph of \(h(x)=f(x) \cdot g(x)\). Where is \(h(x)\) unde ned? How can you indicate this on your graph? How does your graphing calculator deal with the point at which \(h\) is unde ned?

The zeros of the function \(f(x)\) are at \(x=-5,-2,0\), and 5 . Find the zeros of the following functions. If there is not enough information to determine this, say so. (a) \(g(x)=3|f(x)|\) (b) \(h(x)=w(f(x))\), where \(w(x)=-2 x^{2}\) (c) \(p(x)=3 f(x)+1\) (d) \(q(x)=4 f(x+1)\) (e) \(m(x)=4 f(-x)\) (f) \(n(x)=-f(x)\)

Let \(f(x)=|x|\) and \(g(x)=1 / x\). Use your knowledge of the graphs of \(f\) and \(g\) to sketch the graph of \(h(x)=f(x) \cdot g(x)\). Where is \(h(x)\) unde ned? Note: You must deal with the cases \(x>0\) and \(x<0\) separately. This is standard protocol for handling absolute values.

For each of the functions given below, give possible formulas for \(f(x)\) and \(g(x)\) such that \(h(x)=f(g(x))\). Do not let \(g(x)=x\); do not let \(f(x)=x\). (a) \(h(x)=\sqrt{x^{2}+3}\) (b) \(h(x)=\sqrt{x}+\frac{5}{\sqrt{x}}\) (c) \(h(x)=\frac{3}{3 x^{2}+2 x}\) (d) \(h(x)=5\left(x^{2}+3 x^{3}\right)^{3}\)

The graph of \(y=f(x)\) is symmetric about the \(y\) -axis. Which of the following functions is equal to \(f(x)\) ? (a) \(g(x)=-f(x)\) (b) \(h(x)=f(-x)\) (c) \(j(x)=-f(-x)\)

Let \(f(x)=|x|\) and \(g(x)=x\). Use your knowledge of the graphs of \(f\) and \(g\) to sketch the graph of \(h(x)=f(x)+g(x)\).

Let \(j(x)=\frac{2}{3 \sqrt{4 x^{2}+3 x}} .\) Suppose that \(j(x)=h(g(f(x))) .\) Write possible formulas for \(f(x), g(x)\), and \(h(x) .\) None of \(f, g\), and \(h\) should be the identity function.

Let \(j(x)=10\left(x^{-2}+2 x^{2}\right)^{3}\). Give two possible decompositions of \(j(x)\) such that \(j(x)=f(g(h(x)))\). None of the functions \(f, g\), and \(h\) should be the identity function.

Applying what you learned in the last section of this chapter to the pocketful of functions you ve been introduced to (the identity, squaring, reciprocal, and absolute value functions), graph the following functions. Label any asymptotes and \(x\) - and \(y\) -intercepts. (a) \(f(x)=\frac{1}{x^{2}}\) (b) \(g(x)=\left|(x-1)^{2}\right|\) (c) \(h(x)=\left|x^{2}-1\right|\) (d) \(j(x)=\frac{1}{x+1}+2\) (e) \(m(x)=\frac{-1}{x-2}+1\) (f) \(p(x)=\left|\frac{1}{x}\right|\)

Decompose the functions by finding functions \(f(x)\) and \(g(x)\), \(f(x) \neq x\) and \(g(x) \neq x\), such that \(h(x)=f(g(x))\). $$ h(x)=\frac{1}{x^{2}+4} $$

Let \(f(x)=x(x+1), g(x)=x^{3}+2 x^{2}+x\). (a) Simplify the following. i. \(f(x)+g(x)\) ii. \(\frac{f(x)}{g(x)}\) iii. \(\frac{g(x)}{f(x)}\) iv. \(\frac{[f(x)]^{2}}{g(x)}\) (b) Solve \(x f(x)=g(x)\).

Decompose the functions by finding functions \(f(x)\) and \(g(x)\), \(f(x) \neq x\) and \(g(x) \neq x\), such that \(h(x)=f(g(x))\). $$ h(x)=\sqrt{x^{2}+1} $$

The Cambridge Widget Company is producing widgets. The xed costs for the company (costs for rent, equipment, etc.) are $$\$ 20,000.$$ This means that before any widgets are produced, the company must spend $$\$ 20,000.$$ Suppose that each widget produced costs the company an additional $$\$ 10 .$$ Let \(x\) equal the number of widgets the company produced. (a) Write a total cost function, \(C(x)\), that gives the cost of producing \(x\) widgets. (Check that your function works, e.g., check that \(C(1)=20,010\) and \(C(2)=20,020 .)\) Graph \(C(x)\). (b) At what rate is the total cost increasing with the production of each widget? In other words, nd \(\Delta C / \Delta x\). (c) Suppose the company sells widgets for $$\$ 50$$ each. Write a revenue function, \(R(x)\), that tells us the revenue received from selling \(x\) widgets. Graph \(R(x)\). (d) Pro \(\mathrm{t}=\) total revenue \(-\) total cost, so the pro \(\mathrm{t}\) function, \(P(x)\), which tells us the pro \(\mathrm{t}\) the company gets by producing and selling \(x\) widgets, can be found by computing \(R(x)-C(x) .\) Write the pro \(\mathrm{t}\) function and graph it. (e) Find \(P(400)\) and \(P(700)\); interpret your answers. Find \(P(401)\) and \(P(402) .\) By how much does the pro t increase for each additional widget sold? Is \(\Delta P / \Delta x\) constant for all values of \(x\) ? (f) How many widgets must the company sell in order to break even? (Breaking even means that the pro \(\mathrm{t}\) is \(0 ;\) the total cost is equal to the total revenue.) (g) Suppose the Cambridge Widget Company has the equipment to produce at maximum 1200 widgets. Then the domain of the pro \(\mathrm{t}\) function is all integers \(x\) where \(0 \leq x \leq 1200\). What is the range? How many widgets should be produced and sold in order to maximize the company s pro ts?

Decompose the functions by finding functions \(f(x)\) and \(g(x)\), \(f(x) \neq x\) and \(g(x) \neq x\), such that \(h(x)=f(g(x))\). $$ h(x)=(\sqrt{x})^{3}-2 \sqrt{x}+3 $$

Graph the functions in Problems 10 through 18 by starting with the graph of a familiar function and applying appropriate shifts, flips, and stretches. Label all \(x\) - and \(y\) -intercepts and the coordinates of any vertices and corners. Use exact values, not numerical approximations. (a) \(y=(x-1)^{2}\) (b) \(y=-x^{2}-1\)

A photocopying shop has a xed cost of operation of $$\$ 4000$$ per month. In addition, it costs them $$\$ 0.01$$ per page they copy. They charge customers $$\$ 0.07$$ per page. (a) Write a formula for \(R(x)\), the shop s monthly revenue from making \(x\) copies. (b) Write a formula for \(C(x)\), the shop s monthly costs from making \(x\) copies. (c) Write a formula for \(P(x)\), the shop s monthly pro t (or loss if negative) from making \(x\) copies. Pro \(\mathrm{t}\) is computed by subtracting total costs from the total revenue. (d) How many copies must they make per month in order to break even? Breaking even means that the pro \(t\) is zero; the total costs and total revenue are equal. (e) Sketch \(C(x), R(x)\), and \(P(x)\) on the same set of axes and label the break-even point. (f) Find a formula for \(A(x)\), the shop s average cost per copy. (g) Make a table of \(A(x)\) for \(x=0,1,10,100,1000,10000\). (h) Sketch a graph of \(A(x)\).

Find \(f(g(h(x)))\) and \(g(h(f(x)))\). $$ f(x)=\frac{1}{x}, g(x)=\sqrt{x}, h(x)=x-3 $$

Graph the functions by starting with the graph of a familiar function and applying appropriate shifts, flips, and stretches. Label all \(x\) - and \(y\) -intercepts and the coordinates of any vertices and corners. Use exact values, not numerical approximations. (a) \(y=|x+2|\) (b) \(y=-|x|+2\)

Find functions \(f\) and \(g\) such that \(h(x)=f(g(x))\) and neither \(f\) nor \(g\) is the identity function, i.e., \(f(x) \neq x\) and \(g(x) \neq x .\) Answers to these problems are not unique. \(h(x)=3 x^{4}+2 x^{2}+3\)

Find \(f(g(h(x)))\) and \(g(h(f(x)))\). $$ f(x)=3 x^{2}+x, g(x)=x+1, h(x)=\frac{2}{3 x} $$

Graph the functions by starting with the graph of a familiar function and applying appropriate shifts, flips, and stretches. Label all \(x\) - and \(y\) -intercepts and the coordinates of any vertices and corners. Use exact values, not numerical approximations. (a) \(y=-(x+3)^{2}-1\) (b) \(y=(x-3)^{2}+1\)

Find functions \(f\) and \(g\) such that \(h(x)=f(g(x))\) and neither \(f\) nor \(g\) is the identity function, i.e., \(f(x) \neq x\) and \(g(x) \neq x .\) Answers to these problems are not unique. \(h(x)=5(x-\pi)^{2}+4(x-\pi)+7\)

Find \(f(g(h(x)))\) and \(g(h(f(x)))\). $$ f(x)=x+2, g(x)=x^{2}, h(x)=\frac{x}{2-x} $$

Graph the functions by starting with the graph of a familiar function and applying appropriate shifts, flips, and stretches. Label all \(x\) - and \(y\) -intercepts and the coordinates of any vertices and corners. Use exact values, not numerical approximations. (a) \(y=-2(x+1)^{2}+3\) (b) \(y+3=7(x+1)^{2}\)

Find functions \(f\) and \(g\) such that \(h(x)=f(g(x))\) and neither \(f\) nor \(g\) is the identity function, i.e., \(f(x) \neq x\) and \(g(x) \neq x .\) Answers to these problems are not unique. \(h(x)=2|3 x-4|\)

Let \(f(x)=x-3\) and \(g(x)=x^{2}-6 x\). Evaluate and simplify each of the following expressions. (a) \(f(x)+g(x)\) (b) \(f(x)-g(x)\) (c) \(f(x) g(x)\) (d) \(f(g(x))\) (e) \(g(f(x))\) (f) \(\frac{f(x)}{g(x)}\)

Graph the functions by starting with the graph of a familiar function and applying appropriate shifts, flips, and stretches. Label all \(x\) - and \(y\) -intercepts and the coordinates of any vertices and corners. Use exact values, not numerical approximations. (a) \(y=\frac{2}{x+4}+1\) (b) \(y=\frac{-1}{x-\pi}\)

Find functions \(f\) and \(g\) such that \(h(x)=f(g(x))\) and neither \(f\) nor \(g\) is the identity function, i.e., \(f(x) \neq x\) and \(g(x) \neq x .\) Answers to these problems are not unique. \(h(x)=\frac{x+1}{x+2}\)

Let \(f(x)=x-3\) and \(g(x)=x^{2}-6 x\). Find the \(x\) - and \(y\) -intercepts of the following. (a) \(f(x)\) (b) \(g(x)\) (c) \(f(x) g(x)\) (d) \(\frac{f(x)}{g(x)}\)

Graph the functions by starting with the graph of a familiar function and applying appropriate shifts, flips, and stretches. Label all \(x\) - and \(y\) -intercepts and the coordinates of any vertices and corners. Use exact values, not numerical approximations. (a) \(y=-x+\pi\) (b) \(y=-(x+\pi)\)

Find functions \(f\) and \(g\) such that \(h(x)=f(g(x))\) and neither \(f\) nor \(g\) is the identity function, i.e., \(f(x) \neq x\) and \(g(x) \neq x .\) Answers to these problems are not unique. \(h(x)=\sqrt{5 x^{2}+3}\)

How do the \(x\) - and \(y\) -intercepts of \(f(x)\) and \(g(x)\) affect the intercepts of \(f(x) g(x)\) ? State this as a general rule.

Graph the functions by starting with the graph of a familiar function and applying appropriate shifts, flips, and stretches. Label all \(x\) - and \(y\) -intercepts and the coordinates of any vertices and corners. Use exact values, not numerical approximations. (a) \(y-\pi=(x-2 \pi)^{2}\) (b) \(y-\pi=-(x-2 \pi)^{2}\)

Find functions \(f\) and \(g\) such that \(h(x)=f(g(x))\) and neither \(f\) nor \(g\) is the identity function, i.e., \(f(x) \neq x\) and \(g(x) \neq x .\) Answers to these problems are not unique. \(h(x)=3^{2 x}+3^{x}+1\)

Graph the functions by starting with the graph of a familiar function and applying appropriate shifts, flips, and stretches. Label all \(x\) - and \(y\) -intercepts and the coordinates of any vertices and corners. Use exact values, not numerical approximations. (a) \(y=\frac{x+3}{x+2}\) (rewrite \(x+3\) as \(x+2+1\) ) (b) \(y=\frac{x+1}{x-1}\)

How do the \(x\) - and \(y\) -intercepts of \(f(x)\) and \(g(x)\) affect the intercepts of \(\frac{f(x)}{g(x)}\) and the places where \(\frac{f(x)}{g(x)}\) is unde ned?

Graph the functions by starting with the graph of a familiar function and applying appropriate shifts, flips, and stretches. Label all \(x\) - and \(y\) -intercepts and the coordinates of any vertices and corners. Use exact values, not numerical approximations. (a) \(y=-1-2|x+1|\) (b) \(y=-1-2(x+1)^{2}\)

In Problems 18 through 20, find functions \(f, g\), and \(h\) such that \(k(x)=f(g(h(x)))\) and \(f(x) \neq x, g(x) \neq x\), and \(h(x) \neq x\). \(k(x)=\frac{3}{\sqrt{x^{2}+4}}\)

Suppose that the functions \(f, g\), and \(h\) are de ned for all integers. At the top of the following page is a table of some of the values of these functions. $$\begin{array}{lccccccc} \hline x & -2 & -1 & 0 & 1 & 2 & 3 & 4 \\ \hline f(x) & 0 & 2 & 1 & 3 & 4 & -2 & 5 \\ g(x) & 2 & 3 & 4 & 1 & 3 & -1 & 0 \\ h(x) & 3 & 4 & -3 & 2 & 8 & 1 & 2 \\ \hline \end{array}$$ Evaluate the following expressions. If not enough information is available for you to do so, indicate that. (a) \(f(-1) \cdot g(-1)\) (b) \(f(g(-1))\) (c) \(g(f(-1))\) (d) \(h(g(f(2)))\) (e) \(\frac{f(0)+2}{g(0)}\) (f) \(5 h(3)+f(f(1))\) (g) \(f(f(f(0)))\)

For Problems 19 through 21 , let \(f(x)=|x| .\) Graph the functions on the same set of axes. \(g(x)=(x-3)^{2}-4\) and \(f(g(x))\)

Find functions \(f, g\), and \(h\) such that \(k(x)=f(g(h(x)))\) and \(f(x) \neq x, g(x) \neq x\), and \(h(x) \neq x\). \(k(x)=\frac{1}{(\sqrt{x}+1)^{9}}\)

Let \(f(x)=|x| .\) Graph the functions on the same set of axes. \(g(x)=-(x+2)^{2}+1\) and \(f(g(x))\)

Find functions \(f, g\), and \(h\) such that \(k(x)=f(g(h(x)))\) and \(f(x) \neq x, g(x) \neq x\), and \(h(x) \neq x\). \(k(x)=\sqrt{\left(x^{2}+1\right)^{3}+5}\)

You put $$\$ 300$$ in a bank account at \(4 \%\) annual interest compounded annually and you plan to leave it there without making any additional deposits or withdrawals. With each passing year, the amount of money in the account is \(104 \%\) of what it was the previous year. (a) Write a formula for the function \(f\) that takes as input the balance in the account at some particular time and gives as output the balance one year later. Write this formula as one term, not the sum of two terms. (b) Two years after the initial deposit is made, the balance in the account is \(f(f(300))\) and three years after, it is \(f(f(f(300)))\). Explain. (c) What quantity is given by \(f(f(f(f(300))))\) ? (d) Challenge: Write a formula for the function \(g\) that takes as input \(n\), the number of years the deposit of \(M\) dollars has been in the bank, and gives as output the balance in the account.

Let \(f(x)=|x| .\) Graph the functions on the same set of axes. \(g(x)=|x-2|-3\) and \(f(g(x))\)

\(f(x)=\frac{1}{2-x}\) and \(g(x)=x^{2}+1\). Find the following. Simplify your answers. If simplifying is dif cult, consult Appendix A: Algebra. (a) \(2 f(x+1)\) (b) \(f(2 x-2)\) (c) \(g(\sqrt{x}+1)\) (d) \(f(g(x))\) (e) \(g(f(x))\) (f) \(f(f(x))\) (g) \(g\left(\frac{1}{f(x)}\right)\) (h) \(\frac{g(x)}{f(x)}\)

Let \(f(x)=\frac{1}{x}\) and \(g(x)=x^{2} .\) Using what you ve learned in Section \(3.4\), graph the following equations. (a) \(y=f(g(x))\) (b) \(y=|g(x-1)-4|\) (c) \(y=|f(x)|-1\)

Most of the time, when a store provides coupons offering $$\$ 5$$ off any item in the store they include the clause except for sale items. Suppose that clause were omitted and you found an item you wanted on a \(30 \%\) off rack. There would be some ambiguity; should the $$\$ 5$$ be taken off the reduced price, or off the price before the \(30 \%\) discount? Let \(C\) be the function that models the effect of the coupon, \(S\) be the function that models the effect of the sale, and \(x\) be the original price of the item. (a) Which situation corresponds to \(C(S(x))\) ? (b) Which situation corresponds to \(S(C(x))\) ? (c) Which order of composition of the functions is in the buyer s favor?