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Chapter 2: Problem 5

Find the domain of each rational function. $$h(x)=\frac{x+7}{x^{2}-49}$$

Short Answer

Expert verified
The domain of the function \(h(x)=\frac{x+7}{x^{2}-49}\) is all real numbers except \(-7\) and \(7\).

Step by step solution

01

Set the Denominator equal to 0

First, set the denominator of the rational function equal to zero. The equation becomes \(x^{2} - 49 = 0\).
02

Solve for x

Solve the equation for \(x\). This involves moving \(49\) to the other side of the equation, then taking the square root of both sides. This will result in \(x = -7, 7\).
03

Exclude x-values from the Domain

The original function is undefined when \(x\) equals \(-7\) or \(7\), because those x-values cause the denominator to equal zero. Accordingly, these values need to be excluded from the domain of the function. The domain is all real numbers except for \(-7\) and \(7\).

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Most popular questions from this chapter

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.

Use inspection to describe each inequality's solution set. Do not solve any of the inequalities. $$(x-2)^{2}>0$$

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$\frac{2}{x^{2}+3 x+2}-\frac{4}{x^{2}+4 x+3}$$

Will help you prepare for the material covered in the next section. Solve: \(x^{3}+x^{2}=4 x+4\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When solving \(f(x)>0,\) where \(f\) is a polynomial function, I only pay attention to the sign of \(f\) at each test value and not the actual function value.
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