Question
he function \( f \) is defined below. \[ f(x)=\frac{x-4}{x^{2}+3 x-40} \] ind all values of \( x \) that are NOT in the domain of \( f \). f there is more than one value, separate them with commas. \( x=\square \)
Ask by Stuart Salazar. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x=-8,5 \)
Solution
1. The function is defined as
\[
f(x)=\frac{x-4}{x^{2}+3 x-40}
\]
2. To find values that are not in the domain, we must determine when the denominator is zero. Set the denominator equal to zero:
\[
x^{2}+3 x-40=0.
\]
3. Factor the quadratic expression. Look for two numbers that multiply to \(-40\) and add to \(3\). These numbers are \(8\) and \(-5\). Thus, the factorization is:
\[
(x+8)(x-5)=0.
\]
4. Set each factor equal to zero and solve:
\[
x+8=0 \quad \Rightarrow \quad x=-8,
\]
\[
x-5=0 \quad \Rightarrow \quad x=5.
\]
5. Therefore, the values that are not in the domain of \( f \) are \( x=-8 \) and \( x=5 \).
\( x=-8,\,5 \)
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Simplify this solution The Deep Dive
To find the values of \( x \) that are NOT in the domain of the function \( f(x) = \frac{x-4}{x^2 + 3x - 40} \), we need to identify where the denominator is equal to zero, because a division by zero is undefined. First, we can factor the quadratic expression in the denominator: \[ x^2 + 3x - 40 = (x + 8)(x - 5) \] Next, we set the denominator equal to zero to find the values that are excluded from the domain: \[ (x + 8)(x - 5) = 0 \] This gives us two solutions: \[ x + 8 = 0 \quad \Rightarrow \quad x = -8 \] \[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \] Thus, the values of \( x \) that are NOT in the domain of \( f \) are: \( x = -8, 5 \)
