Chapter 2

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$ for the given function. $$f(x)=-x^{2}+2 x+4$$

Begin by graphing the standard quadratic function, \(f(x)-x^{2} .\) Then use transformations of this graph to graph the given function. $$ h(x)--2(x+2)^{2}+1 $$

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$ for the given function. $$f(x)=-x^{2}-3 x+1$$

Begin by graphing the square root function, \(f(x)-\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ g(x)-\sqrt{x}+2 $$

Graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$ \begin{array}{c} x^{2}+y^{2}=16 \\ x-y=4 \end{array} $$

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$ for the given function. $$f(x)=-2 x^{2}+5 x+7$$

Use intercepts to graph each equation. \(6 x-2 y-12=0\)

Begin by graphing the square root function, \(f(x)-\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ g(x)-\sqrt{x}+1 $$

Graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$ \begin{array}{c} x^{2}+y^{2}=9 \\ x-y=3 \end{array} $$

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$ for the given function. $$f(x)=-3 x^{2}+2 x-1$$

Use intercepts to graph each equation. \(6 x-9 y-18-0\)

The formula $$ y=f(x)=\frac{9}{5} x+32 $$ is used to convert from \(x\) degrees Celsius to \(y\) degrees Fahrenheit. The formula $$ y=g(x)=\frac{5}{9}(x-32) $$ is used to convert from \(x\) degrees Fahrenheit to \(y\) degrees Celsius. Show that \(f\) and \(g\) are inverse functions.

Begin by graphing the square root function, \(f(x)-\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ g(x)-\sqrt{x+2} $$

Graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$ \begin{array}{r} (x-2)^{2}+(y+3)^{2}=4 \\ y=x-3 \end{array} $$

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$ for the given function. $$f(x)=-2 x^{2}-x+3$$

Use intercepts to graph each equation. \(2 x+3 y+6-0\)

Explain how to determine if two functions are inverses of each other.

Begin by graphing the square root function, \(f(x)-\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ g(x)-\sqrt{x+1} $$

Graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$ \begin{array}{r} (x-3)^{2}+(y+1)^{2}=9 \\ y=x-1 \end{array} $$

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$ for the given function. $$f(x)=-3 x^{2}+x-1$$

Use intercepts to graph each equation. \(3 x+5 y+15-0\)

Find a. \((f \circ g)(x)\) b. the domain of \(f^{\circ}\) g. $$ f(x)-\frac{x}{x+5}, g(x)=\frac{6}{x} $$

Describe how to find the inverse of a one-to-one function.

Begin by graphing the square root function, \(f(x)-\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ h(x)--\sqrt{x+2} $$

Use intercepts to graph each equation. \(8 x-2 y+12-0\)

What is the horizontal line test and what does it indicate?

Begin by graphing the square root function, \(f(x)-\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ h(x)--\sqrt{x+1} $$

Use intercepts to graph each equation. \(6 x-3 y+15-0\)

Describe how to use the graph of a one-to-one function to draw the graph of its inverse function.

Begin by graphing the square root function, \(f(x)-\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ h(x)-\sqrt{-x+2} $$

Find the slope of the line passing through each pair of points or state that the slope is undefined. Assume that all variables represent positive real numbers. Then indicate whether either line through the points rises, falls, is horizontal, or is vertical. \((0, a)\) and \((b, 0)\)

Find a. \((f \circ g)(x)\) b. the domain of \(f^{\circ}\) g. $$ f(x)-x^{2}+4, g(x)-\sqrt{1-x} $$

How can a graphing utility be used to visually determine if two functions are inverses of each other?

Begin by graphing the square root function, \(f(x)-\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ h(x)-\sqrt{-x+1} $$

Find the slope of the line passing through each pair of points or state that the slope is undefined. Assume that all variables represent positive real numbers. Then indicate whether either line through the points rises, falls, is horizontal, or is vertical. \((-a, 0)\) and \((0,-b)\)

Begin by graphing the square root function, \(f(x)-\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ g(x)-\frac{1}{2} \sqrt{x+2} $$

In your own words, describe how to find the distance between two points in the rectangular coordinate system.

Find the slope of the line passing through each pair of points or state that the slope is undefined. Assume that all variables represent positive real numbers. Then indicate whether either line through the points rises, falls, is horizontal, or is vertical. \((a, b)\) and \((a, b+c)\)

Use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). $$ f(x)=x^{2}-1 $$

Begin by graphing the square root function, \(f(x)-\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ g(x)-2 \sqrt{x+1} $$

In your own words, describe how to find the distance between two points in the rectangular coordinate system.

Find the slope of the line passing through each pair of points or state that the slope is undefined. Assume that all variables represent positive real numbers. Then indicate whether either line through the points rises, falls, is horizontal, or is vertical. \((a-b, c)\) and \((a, a+c)\)

Use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). $$ f(x)=\sqrt[3]{2-x} $$

Begin by graphing the square root function, \(f(x)-\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ h(x)-\sqrt{x+2}-2 $$

What is a circle? Without using variables, describe how the definition of a circle can be used to obtain a form of its equation.

Use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). $$ f(x)=\frac{x^{3}}{2} $$

Begin by graphing the square root function, \(f(x)-\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ h(x)-\sqrt{x+1}-1 $$

Give an example of a circle's equation in standard form. Describe how to find the center and radius for this circle.

Use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). $$ f(x)=\frac{x^{4}}{4} $$

Begin by graphing the square root function, \(f(x)-\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ g(x)-2 \sqrt{x+2}-2 $$