# What is a domain restriction? AND How do you find domain restrictions??

## I understand how to solve for domain restrictions for problems like: $f \left(x\right) = \frac{{x}^{2} - 4}{{x}^{2} - 10 x - 24}$ . But I don't get how to solve problems like: $f \left(x\right) = 120 {\left(\frac{1}{2}\right)}^{x} - 7$ and : $f \left(x\right) = - \sqrt{2 x - 4} + 7$ and also problems like: $f \left(x\right) = 2 {\log}_{4} \left(x\right) - 3$ PLEASE, I NEED HELP WITH DOMAIN RESTRICTIONS!!!!!!

May 4, 2018

Informally, the domain for some function $f \left(x\right)$ consists of all the values of $x$ you are allowed to plug in without "breaking" the rules of math.

For example, consider the function $f \left(x\right) = \frac{1}{x}$. Here, you can plug in every value except $x = 0$, precisely because $\frac{1}{0}$ is not defined. The domain, then, would consist of all values except zero.

When trying to find the domain of a function, it helps to plug in some values in your head as well as consider the parent graph. These both will give you some insight as to what values, if any, need to be excluded from the domain.

You provided the function $f \left(x\right) = 120 {\left(\frac{1}{2}\right)}^{x} - 7$. To discover the domain, ask yourself, "Is there any value of $x$ I can plug in that breaks some math rule"? In this case, there is not. You can raise a constant to any number, positive, negative or zero, and it will still be defined. Thus, this function has a domain of $\left(- \infty , \infty\right)$.

With the function $f \left(x\right) = - \sqrt{2 x - 4} + 7$, we recall that you cannot take the square root of a negative number. Thus, we must have $2 x - 4$ be non-negative. $2 x - 4 \ge 0 \implies 2 x \ge 4 \implies x \ge 2$. Thus, our domain for this function only includes values greater than or equal to 2. That is, the domain is $\left[2 , \infty\right)$.

With the function $f \left(x\right) = 2 {\log}_{4} \left(x\right) - 3$, we recall that you cannot take the logarithm of zero or a negative number. Thus, $x$ cannot be zero or negative. Our domain would be $\left(0 , \infty\right)$.

Be on the lookout for what values of $x$ "break" a function and you will be better able to see where domain restrictions should exist.