Determine the domain of the function. $$ f(x)=\frac{x^{2}}{10-x} $$

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To determine the domain of the function

, we need to identify any values of

that would make the denominator equal to zero, as this would make the function undefined. Setting the denominator equal to zero gives:

Solving for

leads to

. Therefore, the function is undefined at

.

The domain of the function is all real numbers except

. In interval notation, this can be expressed as:

The denominator of the function is $$ 10 - x $$. We need to find when this expression is equal to zero:

$$
10 - x = 0
$$

Solving for $$ x $$:

$$
x = 10
$$

The function $$ f(x) $$ is undefined when $$ x = 10 $$. Therefore, the domain of the function includes all real numbers except $$ x = 10 $$.

In interval notation, the domain of $$ f(x) $$ is:

$$
(-\infty, 10) \cup (10, \infty)
$$

Thus, the domain of the function $$ f(x) = \frac{x^2}{10 - x} $$ is $$ (-\infty, 10) \cup (10, \infty) $$.
The domain of $$ f(x) = \frac{x^2}{10 - x} $$ is $$ (-\infty, 10) \cup (10, \infty) $$.

Determine the domain of the function.

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