Problem 7
Question
Find the domain of each rational function. $$f(x)=\frac{x+7}{x^{2}+49}$$
Step-by-Step Solution
Verified Answer
The domain of the function is all real numbers.
1Step 1: Identify the Denominator
The first step is to identify the denominator of the rational function, which is \(x^{2}+49\).
2Step 2: Solve for x
Setting the denominator equal to zero and solving for \(x\) gives: \[ x^{2}+49=0 \]\[ x^{2}=-49 \]However, there is no real number that you can square to get a negative number, therefore this equation has no real solutions.
3Step 3: Determine the Domain
Since the denominator never equals zero for any real numbers, the domain of the function \(f(x)=\frac{x+7}{x^{2}+49}\) is all real numbers.
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