Bob Delgado
06/03/2024 · Primary School
For the pair of functions \( f \) and \( g \), determine the domain of \( f / g \). \( f(x)=8 x-7, g(x)=7 x-63 \)
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Step-by-step Solution
To determine the domain of the function \( \frac{f}{g} \), we need to find the values of \( x \) for which \( g(x) \neq 0 \), since division by zero is undefined.
Given the functions:
\[
f(x) = 8x - 7
\]
\[
g(x) = 7x - 63
\]
We first set \( g(x) \) equal to zero and solve for \( x \):
\[
7x - 63 = 0
\]
Adding 63 to both sides gives:
\[
7x = 63
\]
Dividing both sides by 7 results in:
\[
x = 9
\]
Thus, \( g(x) = 0 \) when \( x = 9 \). Therefore, the function \( \frac{f}{g} \) is undefined at \( x = 9 \).
The domain of \( \frac{f}{g} \) is all real numbers except \( x = 9 \). In interval notation, this can be expressed as:
\[
(-\infty, 9) \cup (9, \infty)
\]
Thus, the domain of \( \frac{f}{g} \) is:
\[
\boxed{(-\infty, 9) \cup (9, \infty)}
\]
Quick Answer
The domain of \( \frac{f}{g} \) is all real numbers except \( x = 9 \), or \( (-\infty, 9) \cup (9, \infty) \).
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