Chapter 2

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. \((1.8,7.5),(-2.5,2.1)\)

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 . \(f(x)=3-4 x, \quad g(x)=\frac{3-x}{4}\)

Find (a) \((f+g)(x)\), (b) \((f-g)(x)\), (c) \((f g)(x)\), and (d) \((f / g)(x)\). What is the domain of \(f / g\) ? \(f(x)=\sqrt{x^{2}-4}, \quad g(x)=\frac{x^{2}}{x^{2}+1}\)

Use the Vertical Line Test to decide whether \(y\) is a function of \(x\). \(x-y^{2}=0\)

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. \(\\{(a, 1),(b, 2),(c, 3)\\}\)

Sketch the lines through the point with the indicated slopes on the same set of coordinate axes. Point \((-2,-5)\) Slopes (a) \(-1\) (b) \(\frac{3}{4}\) (c) 0 (d) Undefined

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. \((37.5,-12.3),(-6.2,5.9)\)

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 . \(f(x)=x^{3}, \quad g(x)=\sqrt[3]{x}\)

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

Use the Vertical Line Test to decide whether \(y\) is a function of \(x\). \(x^{2}+y^{2}=9\)

Write a linear model that relates the variables. \(H\) varies directly as \(p ; H=27\) when \(p=9\)

Plot the points and find the slope of the line passing through the points. \((6,9),(-4,-1)\)

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. \((-36,-18),(48,-72)\)

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

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. \(\\{(c, 0),(b, 0),(a, 3)\\}\)

Write a linear model that relates the variables. \(s\) is proportional to \(t ; s=32\) when \(t=4\)

Plot the points and find the slope of the line passing through the points. \((2,4),(4,-4)\)

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. \((1.451,3.051),(5.906,11.360)\)

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 . \(f(x)=\sqrt{x-4}, \quad g(x)=x^{2}+4, \quad x \geq 0\)

Evaluate the function for \(f(x)=2 x+1\) and \(g(x)=x^{2}-2\) \((f+g)(3)\)

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

Describe the increasing and decreasing behavior of the function. Find the point or points where the behavior of the function changes. \(f(x)=2 x\)

The domain of \(f\) is the set \(A=\\{-2,-1,0,1,2\\}\) Write the function as a set of ordered pairs. \(f(x)=x^{2}\)

Write a linear model that relates the variables. \(c\) is proportional to \(d ; c=12\) when \(d=20\)

Plot the points and find the slope of the line passing through the points. \((-6,-1),(-6,4)\)

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 . \(f(x)=9-x^{2}, \quad x \geq 0\) \(g(x)=\sqrt{9-x}, \quad x \leq 9\)

Evaluate the function for \(f(x)=2 x+1\) and \(g(x)=x^{2}-2\) \((f-g)(-2)\)

Describe the increasing and decreasing behavior of the function. Find the point or points where the behavior of the function changes. \(f(x)=x^{2}-2 x\)

The domain of \(f\) is the set \(A=\\{-2,-1,0,1,2\\}\) Write the function as a set of ordered pairs. \(f(x)=\frac{2 x}{x^{2}+1}\)

Write a linear model that relates the variables. \(r\) varies directly as \(s ; r=25\) when \(s=40\)

Plot the points and find the slope of the line passing through the points. \((0,-10),(-4,0)\)

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 . \(f(x)=1-x^{3}, \quad g(x)=\sqrt[3]{1-x}\)

Evaluate the function for \(f(x)=2 x+1\) and \(g(x)=x^{2}-2\) \((f-g)(2 t)\)

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

Describe the increasing and decreasing behavior of the function. Find the point or points where the behavior of the function changes. \(f(x)=x^{3}-3 x^{2}\)

The domain of \(f\) is the set \(A=\\{-2,-1,0,1,2\\}\) Write the function as a set of ordered pairs. \(f(x)=\sqrt{x+2}\)

The simple interest received from an investment is directly proportional to the amount of the investment. By investing $$\$ 2500$$ in a bond issue, you obtain an interest payment of $$\$ 187.50$$ at the end of 1 year. Find a mathematical model that gives the interest \(I\) at the end of 1 year in terms of the amount invested \(P\).

Plot the points and find the slope of the line passing through the points. \(\left(-\frac{1}{3}, 1\right),\left(-\frac{2}{3}, \frac{5}{6}\right)\)

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 . \(f(x)=\frac{1}{1+x}, x \geq 0\) \(g(x)=\frac{1-x}{x}, \quad 0

Evaluate the function for \(f(x)=2 x+1\) and \(g(x)=x^{2}-2\) \((f+g)(t-1)\)

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

Describe the increasing and decreasing behavior of the function. Find the point or points where the behavior of the function changes. \(f(x)=\sqrt{x^{2}-4}\)

The domain of \(f\) is the set \(A=\\{-2,-1,0,1,2\\}\) Write the function as a set of ordered pairs. \(f(x)=|x+1|\)

The simple interest received from an investment is directly proportional to the amount of the investment. By investing $$\$ 5000$$ in a municipal bond, you obtain interest of $$\$ 337.50$$ at the end of 1 year. Find a mathematical model that gives the interest \(I\) at the end of 1 year in terms of the amount invested \(P\).

Plot the points and find the slope of the line passing through the points. \(\left(\frac{7}{8}, \frac{3}{4}\right),\left(\frac{5}{4},-\frac{1}{4}\right)\)

Evaluate the function for \(f(x)=2 x+1\) and \(g(x)=x^{2}-2\) \((f g)(-2)\)

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

Describe the increasing and decreasing behavior of the function. Find the point or points where the behavior of the function changes. \(f(x)=3 x^{4}-6 x^{2}\)

Determine whether the equation represents \(y\) as a function of \(x\). \(x^{2}+y^{2}=4\)

Your property tax is based on the assessed value of your property. (The assessed value is often lower than the actual value of the property.) A house that has an assessed value of $$\$ 150,000$$ has a property tax of $$\$ 5520$$. (a) Find a mathematical model that gives the amount of property tax \(y\) in terms of the assessed value \(x\) of the property. (b) Use the model to find the property tax on a house that has an assessed value of \(\$ 185,000\).