Finding the Domain of a Function Define by an Equation Practice

1. $g(x)=\sqrt{x+3}$

   What is the domain of ‍$g$?

   1. All real values of ‍$x$ such that $x\neq -3$
   2. All real values of ‍$x$ such that $x\geq -3$
   3. All real values of ‍$x$ such that $x\neq 3$
   4. All real values of ‍$x$ such that $x\geq 3$

   5. All real values of ‍$x$

2. $h(x)=x-4$

   What is the domain of ‍$h$?

   1. All real values of ‍$x$ such that $x\neq 4$
   2. All real values of ‍$x$ such that $x\geq 4$
   3. All real values of ‍$x$ such that $x\neq 5$
   4. All real values of ‍$x$ such that $x\geq 5$

   5. All real values of ‍$x$

3. $f(x)=\dfrac{x}{x-7}$

   What is the domain of ‍$f$?

   1. All real values of ‍$x$ such that $x\neq 0$
   2. All real values of ‍$x$ such that $x\geq 0$
   3. All real values of ‍$x$ such that $x\neq 7$
   4. All real values of ‍$x$ such that $x\geq 7$

   5. All real values of ‍$x$

4. $h(x)=\sqrt{x-10}$

   What is the domain of ‍$h$?

   1. All real values of ‍$x$ such that $x\neq 0$
   2. All real values of ‍$x$ such that $x\geq 10$
   3. All real values of ‍$x$ such that $x\neq 10$
   4. All real values of ‍$x$ such that $x\geq 0$
   5. All real values of ‍$x$

Source: Khan Academy, https://www.khanacademy.org/math/college-algebra/xa5dd2923c88e7aa8:functions/xa5dd2923c88e7aa8:domain-and-range-of-a-function/e/domain-of-algebraic-functions
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1. $g$ 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 ‍$x\geq -3$.

   ---

2. $g$‍ 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 ‍$x$.

   ---

3. $f$ 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 ‍$x\neq 7$.

   ---

4. $h$ 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 ‍$x\geq 10$.