Finding the Domain of a Function Define by an Equation Practice

$g$ is a square root function.

Square root functions are defined for all real numbers except those which result in a negative expression below the square root.

The expression below the square root in ‍$g(x)=\sqrt{x+3}$ is ‍$x+3$. We want that to be greater than or equal to zero:

$\begin{aligned}x+3&\geq 0\\\\x&\geq -3\end{aligned}$

The domain of ‍$g$ is all real values of ‍$x$ such that ‍$x\geq -3$.

$g$‍ is a linear function.

Is there an input value that would make a linear expression undefined?

There isn't! Linear functions are defined for all real numbers.

The domain of ‍$h$ is all real values of ‍$x$.

$f$ is a rational function.

Rational functions are defined for all real numbers except those which result in a denominator that is equal to zero (i.e., division by zero).

The denominator of $f(x)=\dfrac{x}{x-7}$ is $x-7$. We want that to not be equal to zero:

$\begin{aligned}x-7&\neq 0\\\\x&\neq 7\end{aligned}$

The domain of $f$ is all real values of ‍$x$ such that ‍$x\neq 7$.

$h$ is a square root function.

Square root functions are defined for all real numbers except those which result in a negative expression below the square root.

The expression below the square root in ‍$h(x)=\sqrt{x-10}$ is ‍$x-10$. We want that to be greater than or equal to zero:

$\begin{aligned}x-10&\geq 0\\\\x&\geq 10\end{aligned}$

The domain of ‍$h$ is all real values of ‍$x$ such that ‍$x\geq 10$.