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Chapter 3: Problem 3

Find the domain of each rational function. $$g(x)=\frac{3 x^{2}}{(x-5)(x+4)}$$

Short Answer

Expert verified
The domain of the function is all real numbers except \(x = 5\) and \(x = -4\).

Step by step solution

01

Express the denominator of the function

We first look at the denominator of \(g(x)\), which is \((x-5)(x+4)\). This product equals zero if either of these terms equals zero.
02

Find the x values that make the denominator zero

We set each term equal to zero and solve for x. Setting \(x - 5\) equal to zero and solving for x gives \(x = 5\). Setting \(x + 4\) equal to zero and solving for x gives \(x = -4\). These are the values that make the denominator zero.
03

Exclude these values from the domain

Since the denominator cannot be zero, we exclude these values from the domain. The domain of \(g(x)\) is all real numbers excluding \(x = 5\) and \(x = -4\).

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Most popular questions from this chapter

Solve each inequality in Exercises \(86-91\) using a graphing utility. $$ x^{3}+x^{2}-4 x-4>0 $$

Solve each inequality in Exercises \(65-70\) and graph the solution set on a real number line. $$ \frac{1}{x+1}>\frac{2}{x-1} $$

You have 600 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?

What is a rational inequality?

Exercises will help you prepare for the material covered in the next section. Factor: \(x^{3}+3 x^{2}-x-3\)
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