ðŸŽ‰ Announcing Numerade's $26M Series A, led by IDG Capital!Read how Numerade will revolutionize STEM Learning # Precalculus 7th ## David Cohen, Theodore B. Lee, David Sklar ## Chapter 3 ## Functions ## Educators ### Problem 1 Let$g$denote the minimum wage function represented in Table. $$\begin{array}{|c|c|}\hline\text { Year } & \text { Minimum Wage (dollars) } \\\hline 1975 & 2.00 \\1980 & 3.10 \\1985 & 3.35 \\1990 & 3.80 \\1995 & 4.25 \\2000 & 5.15 \\2005 & 5.15 \\2010 & 7.25 \\\hline\end{array}$$ (a) Find$g(1975)$(b) Find$g(1995)-g(1975)$and interpret the result. CC Charles C. Numerade Educator ### Problem 2 Let$f$denote the function represented in Table. $$\begin{array}{|c|c|}\hline\hline \text { Movie Star } & \text { Original Name } \\\hline \text { Kate Capshaw } & \text { Kathleen Sue Nail } \\\text { Tom Cruise } & \text { Thomas Cruise Mapother, IV } \\\text { Demi Moore } & \text { Demetria Guynes } \\\text { Kevin Spacey } & \text { Kevin Fowler } \\\text { Arnold Schwarzenegger } & \text { Arnold Schwarzenegger }\end{array}$$ (a) Find$f($Kate Capshaw) (b) For which input$x$are$x$and$f(x)$identical? CC Charles C. Numerade Educator ### Problem 3 Let$h$denote the function represented in Table. $$\begin{array}{|c|c|}\hline\hline\text { Planet } & \text { Number of Moons } \\\hline \text { Mercury } & 0 \\\text { Venus } & 0 \\\text { Earth } & 1 \\\text { Mars } & 2 \\\text { Jupiter } & 16 \\\text { Saturn } & 18 \\\text { Uranus } & 18 \\\text { Neptune } & 8\end{array}$$ (a) Is it the domain or the range of$h$that consists of real numbers? (b) Find$h(\text { Mars })$(c) Which is larger:$h$(Neptune) or$h$(Earth)? Interpret the result (using a complete sentence). CC Charles C. Numerade Educator ### Problem 4 Again let$h$denote the function represented in Table 2 . $$\begin{array}{|c|c|}\hline\hline\text { Planet } & \text { Number of Moons } \\\hline \text { Mercury } & 0 \\\text { Venus } & 0 \\\text { Earth } & 1 \\\text { Mars } & 2 \\\text { Jupiter } & 16 \\\text { Saturn } & 18 \\\text { Uranus } & 18 \\\text { Neptune } & 8\end{array}$$ (a) List those inputs$x$for which$h(x)=1$. (b) For which real number$t$will it be true that liJupiter$)+t=h(\text { Saturn }) ?$CC Charles C. Numerade Educator ### Problem 5 Two sets$A$and$B$are defined as follows: $$A=\{x, y, z\} ; B=\{1,2,3\}$$ (a) Which of the rules displayed in the figures represent functions from$A$to$B$? (b) For each rule that does represent a function, specify the range. (FIGURES CAN'T COPY) CC Charles C. Numerade Educator ### Problem 6 Two sets$A$and$B$are defined as follows: $$A=\{x, y, z\} ; B=\{1,2,3\}$$ (a) Which of the rules displayed in the figures represent functions from$A$to$B$? (b) For each rule that does represent a function, specify the range. (FIGURES CAN'T COPY) CC Charles C. Numerade Educator ### Problem 7 Two sets$D$and$C$are defined as follows: $$D=\{a, b\} ; C=\{i, j, k\}$$ (a) Which of the rules displayed in the figures represent functions from$D$to$C$? (b) For each rule that represents a function, specify the range. (FIGURES CAN'T COPY) CC Charles C. Numerade Educator ### Problem 8 Two sets$D$and$C$are defined as follows: $$D=\{a, b\} ; C=\{i, j, k\}$$ (a) Which of the rules displayed in the figures represent functions from$D$to$C$? (b) For each rule that represents a function, specify the range. (FIGURES CAN'T COPY) CC Charles C. Numerade Educator ### Problem 9 Determine the domain of each function. (a)$y=-5 x+1$(b)$y=1 /(-5 x+1)$(c)$y=\sqrt{-5 x+1}$(d)$y=\sqrt[3]{-5 x+1}$CC Charles C. Numerade Educator ### Problem 10 Determine the domain of each function. (a)$s=3 t+12$(b)$s=1 /(3 t+12)$(c)$s=\sqrt{3 t+12}$(d)$s=\sqrt[3]{3 t+12}$CC Charles C. Numerade Educator ### Problem 11 Determine the domain of each function. (a)$f(x)=x^{2}-9$(b)$g(x)=1 /\left(x^{2}-9\right)$(c)$h(x)=\sqrt{x^{2}-9}$(d)$k(x)=\sqrt[3]{x^{2}-9}$CC Charles C. Numerade Educator ### Problem 12 Determine the domain of each function. (a)$F(t)=t^{2}+4 t$(b)$G(t)=1 /\left(t^{2}+4 t\right)$(c)$H(t)=\sqrt{t^{2}+4 t}$(d)$K(t)=\sqrt[3]{t^{2}+4 t}$CC Charles C. Numerade Educator ### Problem 13 Determine the domain of each function. (a)$f(t)=t^{2}-8 t+15$(b)$g(t)=1 /\left(t^{2}-8 t+15\right)$(c)$h(t)=\sqrt{t^{2}-8 t+15}$(d)$k(t)=\sqrt[3]{t^{2}-8 t+15}$CC Charles C. Numerade Educator ### Problem 14 Determine the domain of each function. (a)$F(x)=2 x^{2}+x-6$(b)$G(x)=1 /\left(2 x^{2}+x-6\right)$(c)$H(x)=\sqrt{2 x^{2}+x-6}$(d)$K(x)=\sqrt[3]{2 x^{2}+x-6}$CC Charles C. Numerade Educator ### Problem 15 Determine the domain of each function. (a)$f(x)=(x-2) /(2 x+6)$(b)$g(x)=\sqrt{(x-2) /(2 x+6)}$(c)$h(x)=\sqrt[3]{(x-2) /(2 x+6)}$CC Charles C. Numerade Educator ### Problem 16 Determine the domain of each function. (a)$F(t)=(3 t-4) /(7-2 t)$(b)$G(t)=\sqrt{(3 t-4) /(7-2 t)}$(c)$H(t)=\sqrt[3]{(3 t-4) /(7-2 t)}$CC Charles C. Numerade Educator ### Problem 17 Determine the domain and the range of each function. $$y=4 x-5$$ CC Charles C. Numerade Educator ### Problem 18 Determine the domain and the range of each function. $$y=125-12 x$$ CC Charles C. Numerade Educator ### Problem 19 Determine the domain and the range of each function. $$y=4 x^{3}-5$$ CC Charles C. Numerade Educator ### Problem 20 Determine the domain and the range of each function. $$y=125-12 x^{3}$$ CC Charles C. Numerade Educator ### Problem 21 Determine the domain and the range of each function. $$g(x)=\frac{4 x-20}{3 x-18}$$ CC Charles C. Numerade Educator ### Problem 22 Determine the domain and the range of each function. $$f(x)=\frac{1-x}{x}$$ CC Charles C. Numerade Educator ### Problem 23 Determine the domain and the range of each function. (a)$f(x)=\frac{x+3}{x-5}$(b)$F(x)=\frac{x^{3}+3}{x^{3}-5}$CC Charles C. Numerade Educator ### Problem 24 Determine the domain and the range of each function. (a)$g(x)=\frac{2 x-7}{3 x+24}$(b)$G(x)=\frac{2 x^{3}-7}{3 x^{3}+24}$CC Charles C. Numerade Educator ### Problem 25 Determine the domain and the range of each function. $$s=t^{2}+4$$ CC Charles C. Numerade Educator ### Problem 26 Determine the domain and the range of each function. $$s=2 t^{2}-10$$ CC Charles C. Numerade Educator ### Problem 27 Each of the following rules defines a function with domain the set of all real numbers. Express each rule in the form of an equation. (a) For each real number, subtract 3 and then square the result. (b) For each real number, compute its square and then subtract 3 from the result. (c) For each real number, multiply it by 3 and then square the result. (d) For each real number, compute its square and then multiply the result by 3. CC Charles C. Numerade Educator ### Problem 28 Each of the following rules defines a function with domain equal to the set of all real numbers. Express each rule in words. (a)$y=2 x^{3}+1$(b)$y=2(x+1)^{3}$(c)$y=(2 x+1)^{3}$(d)$y=(2 x)^{3}+1$CC Charles C. Numerade Educator ### Problem 29 Let$f(x)=x^{2}-3 x+1 .$Compute the following. (a)$f(1)$(b)$f(0)$(c)$f(-1)$(d)$f(3 / 2)$(e)$f(z)$(f)$f(x+1)$(g)$f(a+1)$(h)$f(-x)$(i)$|f(1)|$(j)$f(\sqrt{3})$(k)$f(1+\sqrt{2})$(1)$|1-f(2)|$CC Charles C. Numerade Educator ### Problem 30 Let$H(x)=1-x+x^{2}-x^{3}$. (a) Which number is larger,$H(0)$or$H(1) ?$(b) Find$H\left(\frac{1}{2}\right) .$Does$H\left(\frac{1}{2}\right)+H\left(\frac{1}{2}\right)=H(1) ?$CC Charles C. Numerade Educator ### Problem 31 Let$f(x)=3 x^{2} .$Find the following. (a)$f(2 x)$(b)$2 f(x)$(c)$f\left(x^{2}\right)$(d)$[f(x)]^{2}$(e)$f(x / 2)$(f)$f(x) / 2$For checking: No two answers are the same. CC Charles C. Numerade Educator ### Problem 32 Let$f(x)=4-3 x$. Find the following. (a)$f(2)$(b)$f(-3)$(c)$f(2)+f(-3)$(d)$f(2+3)$(e)$f(2 x)$(f)$2 f(x)$(g)$f\left(x^{2}\right)$(h)$f(1 / x)$(i)$f[f(x)]$(j)$x^{2} f(x)$(k)$1 / f(x)$(1)$f(-x)(m)-f(x)$(n)$-f(-x)$CC Charles C. Numerade Educator ### Problem 33 Let$H(x)=1-2 x^{2} .$Find the following. (a)$H(\sqrt{2})$(b)$H(5 / 6)$(c)$H(x+1)$(d)$H(x+h)$CC Charles C. Numerade Educator ### Problem 34 (a) If$f(x)=2 x+1,$does$f(3+1)=f(3)+f(1) ?$(b) If$f(x)=2 x,$does$f(3+1)=f(3)+f(1) ?$(c) If$f(x)=\sqrt{x},$does$f(3+1)=f(3)+f(1) ?$CC Charles C. Numerade Educator ### Problem 35 Let$g(x)=2,$for all$x .$Find each output. (a)$g(0)$(b)$g(5)$(c)$g(x+h)$CC Charles C. Numerade Educator ### Problem 36 Let$g(t)=|t-4| .$Find$g(3) .$Find$g(x+4)$. CC Charles C. Numerade Educator ### Problem 37 Let$f(x)=x^{2}-6 x$. In each case, find all real numbers$x$(if any) that satisfy the given equation. (a)$f(x)=16$(b)$f(x)=-10$(c)$f(x)=-9$CC Charles C. Numerade Educator ### Problem 38 Let$g(t)=(4 t-6) /(t-4) .$In each case, find all the real number solutions (if any) for the given equation. (a)$g(t)=14$(b)$g(t)=4$(c)$g(t)=0$CC Charles C. Numerade Educator ### Problem 39 Refer to the demand function. In each case, solve the indicated equation for$n .$Round each answer to the nearest integer and interpret the result. $$p(n)=8$$ CC Charles C. Numerade Educator ### Problem 40 Refer to the demand function. In each case, solve the indicated equation for$n .$Round each answer to the nearest integer and interpret the result. $$p(n)=18$$ CC Charles C. Numerade Educator ### Problem 41 Refers to the demand function given in Example$7 .$Now assume that a second economist proposes an alternative model f for the demand function: $$f(n)=4+\frac{3000}{n+100} \quad(100 \leq n \leq 1500)$$ where$n$is the number of tee shirts that can be sold per month at a price of$f(n)$dollars per shirt. At a price level of$\$19$ per shirt, compare the predictions for monthly sales obtained using each model.

CC
Charles C.
Refers to the demand function given in Example $7 .$ Now assume that a second economist proposes an alternative model f for the demand function: $$f(n)=4+\frac{3000}{n+100} \quad(100 \leq n \leq 1500)$$
where $n$ is the number of tee shirts that can be sold per month at a price of $f(n)$ dollars per shirt.