Question
Find the domain of the function using interval notation. \[ f(x)=\frac{1}{x^{2}-x-56} \]
Ask by Bond Schofield. in the United States
Mar 10,2025
Upstudy AI Solution
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Answer
The domain of the function \( f(x) = \frac{1}{x^{2} - x - 56} \) is all real numbers except \( x = -7 \) and \( x = 8 \). In interval notation, this is \( (-\infty, -7) \cup (-7, 8) \cup (8, \infty) \).
Solution
1. The denominator of the function is
\[
x^2 - x - 56.
\]
We need to ensure that the denominator is not zero.
2. Factor the quadratic:
\[
x^2 - x - 56 = (x - 8)(x + 7).
\]
3. Set each factor equal to zero to find the critical values:
\[
x - 8 = 0 \quad \Rightarrow \quad x = 8,
\]
\[
x + 7 = 0 \quad \Rightarrow \quad x = -7.
\]
4. These values would make the denominator zero, so they are excluded from the domain.
5. Therefore, the domain using interval notation is:
\[
(-\infty, -7) \cup (-7, 8) \cup (8, \infty).
\]
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Simplify this solution Mind Expander
To find the domain of the function \( f(x) = \frac{1}{x^2 - x - 56} \), we first need to determine where the denominator is not equal to zero, since division by zero is undefined. We set the denominator equal to zero: \[ x^2 - x - 56 = 0 \] Next, we factor the quadratic: \[ (x - 8)(x + 7) = 0 \] This gives us the solutions \( x = 8 \) and \( x = -7 \). Thus, the function \( f(x) \) is undefined at \( x = 8 \) and \( x = -7 \). Now, we can express the domain in interval notation. The function is defined for all real numbers except at these two points: \[ \text{Domain: } (-\infty, -7) \cup (-7, 8) \cup (8, \infty) \]
