Chapter 2

Use translations of one of the basic functions \(y=x^{2}, y=x^{3}\) \(y=\sqrt{x},\) or \(y=|x|\) to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$y=\sqrt{x+2}$$

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand $$f(x)=2 \sqrt{x-2}-1$$

Use translations of one of the basic functions \(y=x^{2}, y=x^{3}\) \(y=\sqrt{x},\) or \(y=|x|\) to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$y=x^{3}+1$$

$$\text {Use transformations of graphs to sketch a graph of } y=f(x) \text { by }$$ $$f(x)=-\sqrt{1-x}$$

Use translations of one of the basic functions \(y=x^{2}, y=x^{3}\) \(y=\sqrt{x},\) or \(y=|x|\) to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$y=|x+2|$$

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. $$f(x)=\sqrt{-x}-1$$

Use a graphing calculator with window \([-5,5]\) by \([-3,3]\) to graph each equation. (Refer to your descriptions in Exercises 41-44.) $$y=-[x]$$

Use translations of one of the basic functions \(y=x^{2}, y=x^{3}\) \(y=\sqrt{x},\) or \(y=|x|\) to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$y=(x-1)^{3}$$

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. $$f(x)=\sqrt{-(x+1)}$$

Use translations of one of the basic functions \(y=x^{2}, y=x^{3}\) \(y=\sqrt{x},\) or \(y=|x|\) to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$y=|x|-3$$

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. $$f(x)=2+\sqrt{-(x-3)}$$

The number of monthly active Whats App users \(W\) was 200 million in 2012 and grew linearly to 700 million by 2015 . It then continued to grow linearly to 1000 million, or 1 billion, by 2018 . (Source: Statista.) (a) Write a formula for a piece wise-linear function \(W(x)\) that models these data, where \(x\) represents the year. (b) Sketch a graph of \(y=W(x) .\) Is \(W\) a continuous function on the interval \([2012,2018] ?\) (c) Interpret the rates of change in \(W\).

Use translations of one of the basic functions \(y=x^{2}, y=x^{3}\) \(y=\sqrt{x},\) or \(y=|x|\) to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$y=\sqrt{x-2}-1$$

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. $$f(x)=(x-1)^{3}$$

The number of daily active Snap chat users \(S\) was 46 million in January 2014 and grew linearly to 94 million by July 2015 . It then continued to grow linearly to 160 million by January 2017 . (Source: Recode.) (a) Write a formula for a piece wise-linear function \(S(x)\) that models these data, where \(x\) represents the number of months after January 2014 . (b) Sketch a graph of \(y=S(x) .\) Is \(S\) a continuous function on the interval \([0,36] ?\) (c) Interpret the rates of change in \(S\).

Use translations of one of the basic functions \(y=x^{2}, y=x^{3}\) \(y=\sqrt{x},\) or \(y=|x|\) to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$y=\sqrt{x+3}-4$$

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. $$f(x)=(x+2)^{3}$$

Use \(f(x)\) and \(g(x)\) to find each composition. Identify is domain. (Use a calculator if necessary to find the domain.) \(\begin{array}{llll}\text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x)\end{array}\) $$f(x)=x^{3}, g(x)=x^{2}+3 x-1$$

Solve each group of equations and inequalities analytically. (a) \(|x+4|=9\) (b) \(|x+4|>9\) (c) \(|x+4|<9\)

Use translations of one of the basic functions \(y=x^{2}, y=x^{3}\) \(y=\sqrt{x},\) or \(y=|x|\) to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$y=(x+2)^{2}+3$$

Use \(f(x)\) and \(g(x)\) to find each composition. Identify is domain. (Use a calculator if necessary to find the domain.) \(\begin{array}{llll}\text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x)\end{array}\) $$f(x)=2-x, g(x)=\frac{1}{x^{2}}$$

Complete the table, assuming that \(g\) is an odd function. $$\begin{array}{|c|c|c|c|c|c|c|c|}\hline x & -5 & -3 & -2 & 0 & 2 & 3 & 5 \\\\\hline g(x) & 13 & & -5 & & & -1 &\end{array}$$

Former professional basketball player Shaquille O'Neal is 7 feet, 1 inch tall and weighs 325 pounds. The table lists his shoe sizes at certain ages. $$\begin{array}{|l|c|c|c|c|} \hline \text { Age } & 20 & 21 & 22 & 23 \\ \hline \text { Shoe Size } & 19 & 20 & 21 & 22 \\\ \hline \end{array}$$ (a) Write a formula that gives his shoe size \(y\) at age \(x=20,21,22,\) and 23 (b) Suppose that after age 23 his shoe size did not change. Sketch a graph of a continuous, piece wise-defined function \(f\) that models his shoe size between the ages 20 and 26, inclusive.

Solve each group of equations and inequalities analytically. (a) \(|x-3|=5\) (b) \(|x-3|>5\) (c) \(|x-3|<5\)

Use translations of one of the basic functions \(y=x^{2}, y=x^{3}\) \(y=\sqrt{x},\) or \(y=|x|\) to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$y=(x-4)^{2}-4$$

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. $$f(x)=(-x)^{3}+1$$

Use \(f(x)\) and \(g(x)\) to find each composition. Identify is domain. (Use a calculator if necessary to find the domain.) \(\begin{array}{llll}\text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x)\end{array}\) $$f(x)=x^{2}, g(x)=\sqrt{1-x}$$

Based on the ordered pairs seen in each table, make a conjecture about whether the finction \(f\) is even, odd, or neither even nor odd. $$\begin{array}{r|r}x & f(x) \\\\\hline-3 & 10 \\\\-2 & 5 \\\\-1 & 2 \\\0 & 1 \\\1 & 2 \\\2 & 5 \\\3 & 10\end{array}$$

In 2017 , the price in dollars for first-class letters weighing up to 5 ounces could be computed by the piece wise constant function \(f,\) where \(x\) is the number of ounces. $$f(x)=\left\\{\begin{array}{ll} 0.49 & \text { if } 0< x \leq 1 \\ 0.70 & \text { if } 1< x \leq 2 \\ 0.91 & \text { if } 2< x \leq 3 \\ 1.12 & \text { if } 3< x \leq 4 \\ 1.33 & \text { if } 4< x \leq 5 \end{array}\right.$$ (a) Evaluate \(f(1.5)\) and \(f(3) .\) Interpret your results. (b) Sketch a graph of \(f .\) What is its domain? What is its range?

Solve each group of equations and inequalities analytically. (a) \(|7-2 x|=3\) (b) \(|7-2 x| \geq 3\) (c) \(|7-2 x| \leq 3\)

Use translations of one of the basic functions \(y=x^{2}, y=x^{3}\) \(y=\sqrt{x},\) or \(y=|x|\) to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$y=|x+4|-2$$

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. $$f(x)=|-x-2|$$

Use \(f(x)\) and \(g(x)\) to find each composition. Identify is domain. (Use a calculator if necessary to find the domain.) \(\begin{array}{llll}\text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x)\end{array}\) $$f(x)=x+2, g(x)=x^{4}+x^{2}-3 x-4$$

The average cost of a 30 -second Super Bowl ad in millions of dollars is approximated by the piece wise-defined function $$f(x)=\left\\{\begin{array}{ll} 0.0475 x-93.3 & \text { if } 1967 \leq x \leq 1998 \\ 0.179 x-356.037 & \text { if } 1998< x \leq 2017 \end{array}\right.$$ where \(x\) represents the year from 1967 to 2017 . Find and interpret each function value. Round values to the nearest hundredth of a million dollars. (b) \(f(1998)\) (a) \(f(1967)\) (c) \(f(2017)\) (d) Is \(f\) continuous on its domain? (e) Graph \(f\).

Based on the ordered pairs seen in each table, make a conjecture about whether the finction \(f\) is even, odd, or neither even nor odd. $$\begin{array}{r|r}x & f(x) \\\\-3 & 10 \\\\-2 & 5 \\\\-1 & 2 \\\0 & 0 \\\1 & -2 \\\2 & -5 \\\3 & -10\end{array}$$

Solve each group of equations and inequalities analytically. (a) \(|-9-3 x|=6\) (b) \(|-9-3 x| \geq 6\) (c) \(|-9-3 x| \leq 6\)

Use translations of one of the basic functions \(y=x^{2}, y=x^{3}\) \(y=\sqrt{x},\) or \(y=|x|\) to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$y=(x+3)^{3}-1$$

Use \(f(x)\) and \(g(x)\) to find each composition. Identify is domain. (Use a calculator if necessary to find the domain.) \(\begin{array}{llll}\text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x)\end{array}\) $$f(x)=\frac{1}{x+1}, g(x)=5 x$$

Based on the ordered pairs seen in each table, make a conjecture about whether the finction \(f\) is even, odd, or neither even nor odd. $$\begin{array}{r|r}x & f(x) \\\\\hline-3 & 10 \\\\-2 & 5 \\\\-1 & 2 \\\0 & 0 \\\1 & -2 \\\2 & -5 \\\3 & -10\end{array}$$

Using interval notation, the table lists the numbers of victims of violent crime per 1000 people for a recent year by age group. $$\begin{array}{|c|c|} \hline \text { Age } & \text { Crime Rate } \\ \hline [12,15) & 28 \\ [15,18) & 23 \\ [18,21) & 34 \\ [21,25) & 27 \\ [25,35) & 19 \\ [35,50) & 13 \\ [50,65) & 11 \\ [65,90) & 2 \\ \hline \end{array}$$ (a) Sketch the graph of a piece wise-defined function that models the data, where \(x\) represents age. (b) Discuss the impact that age has on the likelihood of being a victim of a violent crime.

Solve each group of equations and inequalities analytically. (a) \(|2 x+1|+3=5\) (b) \(|2 x+1|+3 \leq 5\) (c) \(|2 x+1|+3 \geq 5\)

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. $$f(x)=|-x+3|$$

Using interval notation, the table lists the numbers in millions of houses built for various time intervals from 1950 to 2020 . $$\begin{array}{|c|c|} \hline \text { Year } & \text { Houses } \\ \hline [1950,1960) & 13.6 \\ [1960,1970) & 16.1 \\ [1970,1980) & 11.6 \\ [1980,1990) & 9.9 \\ [1990,2000) & 11.0 \\ [2000,2010) & 15.4 \\ [2010,2020) & 12.9^{*} \\ \hline \end{array}$$ (a) Sketch the graph of a piecewise-defined function that models the data, where \(x\) represents the year. (b) Discuss the trends in housing starts between 1950 and 2020.

Based on the ordered pairs seen in each table, make a conjecture about whether the finction \(f\) is even, odd, or neither even nor odd. $$\begin{array}{r|r}x & f(x) \\\\-3 & -5 \\\\-2 & -4 \\\\-1 & -1 \\\0 & 0 \\\1 & 1 \\\2 & 4 \\\3 & 5\end{array}$$

Solve each group of equations and inequalities analytically. (a) \(|4 x+7|+4=4\) (b) \(|4 x+7|+4>4\) (c) \(|4 x+7|+4<4\)

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. $$f(x)=|-2 x+1|$$

Use \(f(x)\) and \(g(x)\) to find each composition. Identify is domain. (Use a calculator if necessary to find the domain.) \(\begin{array}{llll}\text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x)\end{array}\) $$f(x)=2 x+1, g(x)=4 x^{3}-5 x^{2}$$

Based on the ordered pairs seen in each table, make a conjecture about whether the finction \(f\) is even, odd, or neither even nor odd. $$\begin{array}{r|r}x & f(x) \\\\-3 & 5 \\\\-2 & 4 \\\\-1 & 3 \\\0 & 2 \\\1 & 1 \\\2 & 0 \\\3 & -1\end{array}$$

Solve each group of equations and inequalities analytically. (a) \(|5-7 x|=0\) (b) \(|5-7 x| \geq 0\) (c) \(|5-7 x| \leq 0\)

Suppose that hand k are both positive numbers. Match each equation with the correct graph in choices A-D. (Graph cannot copy) $$y=(x+h)^{2}+k$$