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Chapter 2: Problem 4

Find the domain of each rational function. $$g(x)=\frac{2 x^{2}}{(x-2)(x+6)}$$

Short Answer

Expert verified
The domain of the function \(g(x)=\frac{2 x^{2}}{(x-2)(x+6)}\) is \((-\infty,-6) \cup (-6,2) \cup (2,\infty)\).

Step by step solution

01

Identifying the Function's Denominator

Identify the denominator of the rational function, which is \((x-2)(x+6)\). The denominator cannot be equal to zero.
02

Set the Denominator Equals to Zero

Set the denominator equal to zero and solve for \(x\), i.e., solve the equation \((x-2)(x+6)=0\) for \(x\).
03

Solving for x-values

This equation is a quadratic equation. Set each factor of the equation equal to zero to solve for \(x\), that's giving two equations \(x-2=0\) and \(x+6=0\). Solving these two equations gives \(x=2\) and \(x=-6\).
04

Expressing the Domain of the Function

The domain of the function is all real numbers except for the values identified, \(x 鈮� 2\) and \(x 鈮� -6\). Hence, the domain of \(g(x)\) in interval notation is \((-\infty,-6) \cup (-6,2) \cup (2,\infty)\).

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Most popular questions from this chapter

Solve each inequality using a graphing utility. $$x^{3}+x^{2}-4 x-4>0$$

Find the horizontal asymptote, if there is one, of the graph of rational function. $$f(x)=\frac{-2 x+1}{3 x+5}$$

Use transformations of \(f(x)=\frac{1}{x}\) or \(f(x)=\frac{1}{x^{2}}\) to graph each rational function. $$h(x)=\frac{1}{(x-3)^{2}}+2$$

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 \(y=\frac{1}{x^{n}} ?\)

a. Find the slant asymptote of the graph of each rational function and \(\mathbf{b} .\) Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}-1}{x}$$
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