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Chapter 3: Problem 7

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

Short Answer

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

Step by step solution

01

Identify the Denominator

The denominator of the rational function is \(x^{2}+49\). We need to find all the values of x for which this is not equal to zero.
02

Set the Denominator Equal to Zero

To find the values that are not in the domain of the function, solve the equation \(x^{2}+49 = 0\) for x.
03

Solve the Equation

Subtract 49 from both sides of the equation to get \(x^{2} = -49\). As x虏 is equal to a negative number, it means there is no real number whose square is negative. Hence, there are no real solutions to this equation.
04

Establish the Domain

Since there are no real values of x that can make the denominator zero, the denominator of the function is always non-zero. Hence, the domain of the function, f(x), is all real numbers.

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