Question

determine the domain of the function $f(x) = -dfrac{7}{x^2 + 9x}$. enter your answer in interval notation.
domain

Explanation:

Step1: Identify the denominator

The function is \( f(x) = -\frac{7}{x^2 + 9x} \). The denominator is \( x^2 + 9x \).

Step2: Set denominator not equal to zero

To find the domain, we need to ensure the denominator is not zero. So, solve \( x^2 + 9x
eq 0 \).
Factor the left side: \( x(x + 9)
eq 0 \).
This means \( x
eq 0 \) and \( x + 9
eq 0 \) (i.e., \( x
eq -9 \)).

Step3: Write the domain in interval notation

The domain is all real numbers except \( x = -9 \) and \( x = 0 \). In interval notation, this is \( (-\infty, -9) \cup (-9, 0) \cup (0, \infty) \).

Answer:

\( (-\infty, -9) \cup (-9, 0) \cup (0, \infty) \)