Question
Domain of the function f(x)=x+6/(x-7)(x+5)
Answers
GPT-4o mini
To find the domain of the function \( f(x) = \frac{x + 6}{(x - 7)(x + 5)} \), we need to identify the values of \( x \) for which the function is defined. The function will be undefined where the denominator equals zero.
1. Set the denominator equal to zero:
\[
(x - 7)(x + 5) = 0
\]
2. Solve for \( x \):
- \( x - 7 = 0 \) gives \( x = 7 \)
- \( x + 5 = 0 \) gives \( x = -5 \)
Therefore, the function \( f(x) \) is undefined at \( x = 7 \) and \( x = -5 \).
3. The domain of the function is all real numbers except \( x = 7 \) and \( x = -5 \). In interval notation, the domain can be expressed as:
\[
(-\infty, -5) \cup (-5, 7) \cup (7, \infty)
\]
Thus, the domain of the function \( f(x) \) is \( (-\infty, -5) \cup (-5, 7) \cup (7, \infty) \).
1. Set the denominator equal to zero:
\[
(x - 7)(x + 5) = 0
\]
2. Solve for \( x \):
- \( x - 7 = 0 \) gives \( x = 7 \)
- \( x + 5 = 0 \) gives \( x = -5 \)
Therefore, the function \( f(x) \) is undefined at \( x = 7 \) and \( x = -5 \).
3. The domain of the function is all real numbers except \( x = 7 \) and \( x = -5 \). In interval notation, the domain can be expressed as:
\[
(-\infty, -5) \cup (-5, 7) \cup (7, \infty)
\]
Thus, the domain of the function \( f(x) \) is \( (-\infty, -5) \cup (-5, 7) \cup (7, \infty) \).