To find the domain of the function [tex]\( f(x) = \frac{x^2 - 16}{x - 4} \)[/tex], we should first look for any values of [tex]\( x \)[/tex] that would make the denominator zero, as these would cause the function to be undefined.
1. Identify the points where the function is undefined:
The denominator of our function is [tex]\( x - 4 \)[/tex]. Set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
x - 4 = 0 \implies x = 4
\][/tex]
So, the function is undefined at [tex]\( x = 4 \)[/tex].
2. Simplify the function if possible:
Next, simplify the numerator [tex]\( x^2 - 16 \)[/tex]. Notice that the numerator can be factored using the difference of squares:
[tex]\[
x^2 - 16 = (x - 4)(x + 4)
\][/tex]
So, our function can be rewritten as:
[tex]\[
f(x) = \frac{(x - 4)(x + 4)}{x - 4}
\][/tex]
For [tex]\( x \neq 4 \)[/tex], the [tex]\( x - 4 \)[/tex] terms cancel out, leaving:
[tex]\[
f(x) = x + 4 \text{ for } x \neq 4
\][/tex]
Although we have simplified the function, we must remember that the original function was undefined at [tex]\( x = 4 \)[/tex].
3. Write the domain in interval notation:
Since the function is undefined at [tex]\( x = 4 \)[/tex], we exclude this value from the domain. Thus, the domain includes all real numbers except [tex]\( x = 4 \)[/tex].
In interval notation, the domain is:
[tex]\[
(-\infty, 4) \cup (4, \infty)
\][/tex]
Thus, the domain of [tex]\( f(x) \)[/tex] is [tex]\(\boxed{(-\infty, 4) \cup (4, \infty)}\)[/tex].
