Chapter 2

Find the inverse function of the function \(f\) given by the set of ordered pairs. \(\\{(1,4),(2,5),(3,6),(4,7)\\}\)

Describe the sequence of transformations from \(f(x)=x^{2}\) to \(g\). Then sketch the graph of \(g\) by hand. Verify with a graphing utility. \(g(x)=x^{2}-4\)

Find the domain and range of the function. Then evaluate \(f\) at the given \(x\) -value. \(f(x)=\sqrt{x-1}\) \(x=1\)

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. \((2,-5),(-6,1)\)

Find the inverse function of the function \(f\) given by the set of ordered pairs. \(\\{(6,2),(5,3),(4,4),(3,5)\\}\)

Describe the sequence of transformations from \(f(x)=x^{2}\) to \(g\). Then sketch the graph of \(g\) by hand. Verify with a graphing utility. \(g(x)=x^{2}+1\)

Find the domain and range of the function. Then evaluate \(f\) at the given \(x\) -value. \(f(x)=\sqrt{x^{2}-4}\) \(x=-2\)

The numbers of working-age civilians (in millions) in the United States that were not involved in the labor force from 1995 to 2005 are given by the following ordered pairs. \((1995,66.3) \quad(1996,66.6) \quad(1997,66.8)\) \((1998,67.5) \quad(1999,68.4)\) \((2000,70.0)\) \(\begin{array}{lll}(2001,71.4) & (2002,72.7) & (2003,74.7)\end{array}\) \((2004,76.0) \quad(2005,76.8)\) A linear model that approximates the data is \(y=1.16 t+59.1,5 \leq t \leq 15\), where \(y\) is the number of civilians (in millions) and \(t=5\) represents \(1995 .\) Plot the actual data with the model. How closely does the model represent the data? (Source: U.S. Bureau of Labor Statistics)

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. \((1,12),(6,0)\)

Find the inverse function of the function \(f\) given by the set of ordered pairs. \(\\{(-1,1),(-2,2),(-3,3),(-4,4)\\}\)

Describe the sequence of transformations from \(f(x)=x^{2}\) to \(g\). Then sketch the graph of \(g\) by hand. Verify with a graphing utility. \(g(x)=(x+2)^{2}\)

Find the domain and range of the function. Then evaluate \(f\) at the given \(x\) -value. \(f(x)=4-x^{2}, x=0\)

The yearly revenues (in billions of dollars) of UPS from 1997 to 2005 are given by the following ordered pairs. \(\begin{array}{lll}(1997,22.5) & (1998,24.8) & (1999,27.1) \\ (2000,29.8) & (2001,30.6) & (2002,31.3) \\\ (2003,33.5) & (2004,36.6) & (2005,42.6)\end{array}\) Use a graphing utility to create a scatter plot of the data. Let \(x=7\) represent 1997 . Then use the regression feature of the graphing utility to find a best-fitting line for the data. Graph the model and the data together. How closely does the model represent the data? (Source: United Parcel Service)

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. \((3,-11),(-12,-3)\)

Find the inverse function of the function \(f\) given by the set of ordered pairs. \(\\{(6,-2),(5,-3),(4,-4),(3,-5)\\}\)

Describe the sequence of transformations from \(f(x)=x^{2}\) to \(g\). Then sketch the graph of \(g\) by hand. Verify with a graphing utility. \(g(x)=(x-3)^{2}\)

Find the domain and range of the function. Then evaluate \(f\) at the given \(x\) -value. \(f(x)=|x-2|, x=2\)

For urban consumers of educational and communication materials, the Consumer Price Index giving the dollar amount equal to the buying power of \(\$ 100\) in December 1997 is given for each year from 1994 to 2005 by the following ordered pairs. \((1996,95.3)\) \((1994,88.8)\) \((1997,98.4)\) \((2000,102.5)\) \((1995,92.2)\) 8 \((1998,100.3)\) \((2002,107.9)\) \((1999,101.2)\) \((2001,105.2)\) \(\begin{array}{lll}(2003,109.8) & (2004,111.6) & (2005,113.7)\end{array}\) Use a graphing utility to create a scatter plot of the data. Let \(x=4\) represent 1994 . Then use the regression feature of the graphing utility to find a best-fitting line for the data. Graph the model and the data together. How closely does the model represent the data? (Source: U.S. Bureau of Labor Statistics

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. \((-7,3),(2,-9)\)

In Exercises \(5-8\), find the inverse function informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). \(f(x)=2 x\)

Find (a) \((f+g)(x)\), (b) \((f-g)(x)\), (c) \((f g)(x)\), and (d) \((f / g)(x)\). What is the domain of \(f / g\) ? \(f(x)=x+1, \quad g(x)=x-1\)

Describe the sequence of transformations from \(f(x)=x^{2}\) to \(g\). Then sketch the graph of \(g\) by hand. Verify with a graphing utility. \(g(x)=(x-2)^{2}+2\)

Find the domain and range of the function. Then evaluate \(f\) at the given \(x\) -value. \(f(x)=x^{3}-1, x=0\)

Decide whether the set of ordered pairs represents a function from \(A\) to \(B\). \(A=\\{0,1,2,3\\}\) and \(B=\\{-2,-1,0,1,2\\}\) Give reasons for your answers. \(\\{(0,1),(1,-2),(2,0),(3,2)\\}\)

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. \((-1,2),(5,4)\)

In Exercises \(5-8\), find the inverse function informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). \(f(x)=-\frac{x}{4}\)

Find (a) \((f+g)(x)\), (b) \((f-g)(x)\), (c) \((f g)(x)\), and (d) \((f / g)(x)\). What is the domain of \(f / g\) ? \(f(x)=2 x-3, g(x)=1-x\)

Describe the sequence of transformations from \(f(x)=x^{2}\) to \(g\). Then sketch the graph of \(g\) by hand. Verify with a graphing utility. \(g(x)=(x+1)^{2}-3\)

Find the domain and range of the function. Then evaluate \(f\) at the given \(x\) -value. \(f(x)=\frac{|x|}{x}, x=5\)

Decide whether the set of ordered pairs represents a function from \(A\) to \(B\). \(A=\\{0,1,2,3\\}\) and \(B=\\{-2,-1,0,1,2\\}\) Give reasons for your answers. \(\\{(0,-1),(2,2),(1,-2),(3,0),(1,1)\\}\)

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. \((2,10),(10,2)\)

In Exercises \(5-8\), find the inverse function informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). \(f(x)=x-5\)

Find (a) \((f+g)(x)\), (b) \((f-g)(x)\), (c) \((f g)(x)\), and (d) \((f / g)(x)\). What is the domain of \(f / g\) ? \(f(x)=x^{2}, g(x)=1-x\)

Describe the sequence of transformations from \(f(x)=x^{2}\) to \(g\). Then sketch the graph of \(g\) by hand. Verify with a graphing utility. \(g(x)=-x^{2}+1\)

Find the domain and range of the function. Then evaluate \(f\) at the given \(x\) -value. \(f(x)=\sqrt{25-x^{2}}\) \(x=0\)

Decide whether the set of ordered pairs represents a function from \(A\) to \(B\). \(A=\\{0,1,2,3\\}\) and \(B=\\{-2,-1,0,1,2\\}\) Give reasons for your answers. \(\\{(0,0),(1,0),(2,0),(3,0)\\}\)

Determine if a line with the following description has a positive slope, a negative slope, or an undefined slope. Line rises from left to right

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. \(\left(\frac{1}{2}, 1\right),\left(-\frac{5}{2}, \frac{4}{3}\right)\)

In Exercises \(5-8\), find the inverse function informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). \(f(x)=x+7\)

Find (a) \((f+g)(x)\), (b) \((f-g)(x)\), (c) \((f g)(x)\), and (d) \((f / g)(x)\). What is the domain of \(f / g\) ? \(f(x)=2 x+3, \quad g(x)=x^{2}-1\)

Find the domain and range of the function. Then evaluate \(f\) at the given \(x\) -value. \(f(x)=\sqrt{x^{2}-9}\) \(x=3\)

Decide whether the set of ordered pairs represents a function from \(A\) to \(B\). \(A=\\{0,1,2,3\\}\) and \(B=\\{-2,-1,0,1,2\\}\) Give reasons for your answers. \(\\{(0,2),(3,0),(1,1)\\}\)

Determine if a line with the following description has a positive slope, a negative slope, or an undefined slope. Vertical line

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. \(\left(-\frac{1}{3},-\frac{1}{3}\right),\left(-\frac{1}{6},-\frac{1}{2}\right)\)

Show that \(f\) and \(g\) are inverse functions by (a) using the definition of inverse functions and (b) graphing the functions. Make sure you test a few points, as shown in Examples 6 and 7 . \(f(x)=5 x+1, \quad g(x)=\frac{x-1}{5}\)

Find (a) \((f+g)(x)\), (b) \((f-g)(x)\), (c) \((f g)(x)\), and (d) \((f / g)(x)\). What is the domain of \(f / g\) ? \(f(x)=x^{2}+5, \quad g(x)=\sqrt{1-x}\)

Describe the sequence of transformations from \(f(x)=|x|\) to \(g .\) Then sketch the graph of \(g\) by hand. Verify with a graphing utility. \(g(x)=|x|+2\)

Use the Vertical Line Test to decide whether \(y\) is a function of \(x\). \(y=x^{2}\)

Decide whether the set of ordered pairs represents a function from \(A\) to \(B\). \(A=\\{a, b, c\\}\) and \(B=\\{0,1,2,3\\}\) Give reasons for your answers. \(\\{(a, 1),(c, 2),(c, 3),(b, 3)\\}\)

Sketch the lines through the point with the indicated slopes on the same set of coordinate axes. Point \((-3,4)\) Slopes (a) \(-2\) (b) \(\frac{2}{3}\) (c) 0 (d) Undefined