To determine the domain of the function [tex]\( f(x) = \frac{x^2}{3 - x} \)[/tex], we need to identify all the values of [tex]\( x \)[/tex] for which the function is defined.
1. Identify restrictions on the domain:
The function [tex]\( f(x) \)[/tex] is a rational function, which means it is defined for all real numbers except where the denominator is zero. The denominator of our function is [tex]\( 3 - x \)[/tex].
2. Solve for the values that make the denominator zero:
To find the values that make the denominator zero, we set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
3 - x = 0
\][/tex]
Solving for [tex]\( x \)[/tex], we get:
[tex]\[
x = 3
\][/tex]
3. Determine the domain:
Since the function is undefined when [tex]\( x = 3 \)[/tex], we exclude this value from the domain. Thus, the domain includes all real numbers except [tex]\( x = 3 \)[/tex].
4. Express the domain in interval notation:
To express this in interval notation, we write:
[tex]\[
(-\infty, 3) \cup (3, \infty)
\][/tex]
So, the domain of the function [tex]\( f(x) = \frac{x^2}{3 - x} \)[/tex] is
[tex]\[
\boxed{(-\infty, 3) \cup (3, \infty)}
\][/tex]
