Question
Given \( f(x)=\frac{x^{2}+x-30}{x-5} \), Part 1 of 5 (a) Write the domain of \( f \) in interval notation. The domain of the function in interval notation is
Ask by Lindsey Gough. in the United States
Mar 17,2025
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Answer
The domain of \( f(x) \) is \( (-\infty, 5) \cup (5, \infty) \).
Solution
To determine the domain of the function \( f(x) = \frac{x^{2} + x - 30}{x - 5} \), we need to identify any values of \( x \) that would make the denominator equal to zero, as these values are not included in the domain.
1. **Identify the denominator**: The denominator is \( x - 5 \).
2. **Set the denominator to zero**:
\[
x - 5 = 0
\]
Solving this gives:
\[
x = 5
\]
3. **Determine the domain**: The function is defined for all real numbers except \( x = 5 \). Therefore, the domain in interval notation excludes this point.
The domain of the function in interval notation is:
\[
(-\infty, 5) \cup (5, \infty)
\]
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Simplify this solution Extra Insights
The domain of the function \( f(x) = \frac{x^{2} + x - 30}{x - 5} \) excludes the value that makes the denominator zero. Setting the denominator equal to zero gives \( x - 5 = 0 \), hence \( x = 5 \). Thus, the domain of \( f \) is all real numbers except 5. In interval notation, this is written as \( (-\infty, 5) \cup (5, \infty) \). To ensure clarity, always remember that the point which causes a division by zero cannot be included in your domain. Keeping track of these restrictions helps define where your function is valid!
