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 \(f(x) = \frac{x+7}{x^{2}+49}\) is all real numbers.
1Step 1: 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.
2Step 2: 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.
3Step 3: Solve the Equation
Subtract 49 from both sides of the equation to get \(x^{2} = -49\). As x² 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.
4Step 4: 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.
Other exercises in this chapter
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