Chapter 3

Exercises will help you prepare for the material covered in the next section. Determine whether \(f(x)=x^{4}-2 x^{2}+1\) is even, odd, or neither. Describe the symmetry, if any, for the graph of \(f\)

In Exercises \(104-107\), use inspection to describe each inequality's solution set. Do not solve any of the inequalities. $$ (x-2)^{2} \leq 0 $$

In Exercises \(104-107\), use inspection to describe each inequality's solution set. Do not solve any of the inequalities. $$ \frac{1}{(x-2)^{2}}>0 $$

What is a rational function?

Use everyday language to describe the behavior of a graph near its vertical asymptote if \(f(x) \rightarrow \infty\) as \(x \rightarrow-2^{-}\) and \(f(x) \rightarrow-\infty\) as \(x \rightarrow-2^{+}\)

If you are given the equation of a rational function, explain how to find the vertical asymptotes, if there is one, of the function's graph.

If you are given the equation of a rational function, explain how to find the horizontal asymptote, if any, of the function's graph.

Exercises \(110-112\) will help you prepare for the material covered in the next section. If \(S-\frac{k A}{P}\), find the value of \(k\) using \(A-60,000, P-40\) and \(S-12.000\)

Describe how to graph a rational function.

If you are given the equation of a rational function, how can you tell if the graph has a slant asymptote? If it does, how do you find its equation?

Is every rational function a polynomial function? Why or why not? Does a true statement result if the two adjectives rational and polynomial are reversed? Explain.

Use a graphing utility to graph \(y-\frac{1}{x}, y-\frac{1}{x^{3}},\) and \(\frac{1}{x^{5}}\) in the same viewing rectangle. For odd values of \(n,\) how does changing \(n\) affect the graph of \(y-\frac{1}{x^{n}} ?\)

Use a graphing utility to graph \(y-\frac{1}{x^{2}}, y-\frac{1}{x^{4}},\) and \(y-\frac{1}{x^{6}}\) in the same viewing rectangle. For even values of \(n,\) how does changing \(n\) affect the graph of \(y-\frac{1}{x^{n}} ?\)

Use a graphing utility to graph $$f(x)-\frac{x^{2}-4 x+3}{x-2} \text { and } g(x)-\frac{x^{2}-5 x+6}{x-2}$$ What differences do you observe between the graph of \(f\) and the graph of \(g\) ? How do you account for these differences?

The rational function $$f(x)-\frac{27,725(x-14)}{x^{2}+9}-5 x$$ models the number of arrests, \(f(x)\), per \(100,000\) drivers, for driving under the influence of alcohol, as a function of a driver's age, \(x\). a. Graph the function in a \([0,70,5]\) by \([0,400,20]\) viewing rectangle. b. Describe the trend shown by the graph. c. Use the ZOOM and TRACE features or the the maximum function feature of your graphing utility to find the age that corresponds to the greatest number of arrests. How many arrests, per \(100,000\) drivers, are there for this age group?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. As production level increases, the average cost for a company to produce each unit of its product also increases.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. It is possible to have a rational function whose graph has no \(y\) -intercept.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of a rational function can have three vertical asymptotes.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of a rational function can never cross a vertical asymptote.

Write the equation of a rational function \(f(x)-\frac{p(x)}{q(x)}\) having the indicated properties, in which the degrees of \(p\) and \(q\) are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. \(f\) has a vertical asymptote given by \(x-3,\) a horizontal asymptote \(y-0, y\) -intercept at \(-1,\) and no \(x\) -intercept.

This will help you prepare for the material covered in the next section. $$\text { Solve: } x^{3}+x^{2}-4 x+4$$

This will help you prepare for the material covered in the next section. $$\text { Simplify: } \frac{x+1}{x+3}-2$$