High School Math Solutions – Functions Calculator, Domain

Determining the domain of a function by looking at its graph is easy to do. However, when you don’t have a graph and have to determine the domain using only the function, it becomes a little harder. In this blog post, we will talk about how to determine the domain of linear, quadratic, radical, and rational functions.

The domain of a function is the set of input or argument values for which the function is real and defined. In simpler terms, the domain is all the values of x that we can plug into the function, keeping the function real and defined. Some examples of values of x that aren’t in the domain are the values of x that cause the function to output an imaginary number or the values of x that cause a zero in the denominator.

Linear and quadratic functions -

Linear and quadratic functions do not have any restrictions on their domain because they are defined and real on all the values of x, unless it’s a piecewise function. That means the domain is -∞<x<∞, which can also be denoted as (-∞,∞).

For example (click here):
                                                     Find the domain of \f(x)=x^3+5
                                                      Domain: -∞<x<∞ or (-∞,∞)
The function is defined and real on all the values of x.

Radical and rational functions -

Radical and rational functions, however, do have restrictions on their domains. Radical functions have a restriction on their domain when the expression under the radical is negative. Rational functions have a restriction on their domain whenever the denominator equals zero.

Let’s see the steps to determine the domain.

Determining the domain of radical functions:

1. Set the expression inside the radical greater than or equal to 0

• This is will find the domain, for which the function is real

2. Solve for x

Determining the domain of rational functions:

1. Set the expression in the denominator equal to 0

• This will find the values, for which the function is undefined.

2. Solve for x



Let’s see some examples.

Here’s an example for radical functions (click here):

                                               Find the domain of \f(x)=5\sqrt{x^2-9}

1. Set the expression inside the radical greater than or equal to 0

                                                                x^2-9≥0

2. Solve for x

                                                                x^2-9≥0

                                                                 x^2≥9

                                                     x≥3     or  x≤-3        (*)

                                           Domain:x≥3 or  x≤-3,or (-∞,-3)∪(3,∞)

(*) If \f(x)^2≥a,then \f(x)≥\sqrt{a}  and \f(x)≤-\sqrt{a}


Here’s an example for rational functions (click here):

                                            Find the domain of \y=\frac{x}{x^2-6x+8}

1. Set the denominator equal to 0

                                                           x^2-6x+8=0

2. Solve for x

                                                           x^2-6x+8=0

                                                           (x-4)(x-2)=0

                                                            x=2 or x=4

                            Domain:x<2 or  2<x<4 or x>4 ,or (-∞,2)∪(2,4)∪(4,∞)

Note: The domain is the values of x which do not equal 2 or 4, because when x=2 or x=4, the function is undefined (i.e. it causes the denominator to equal 0.

Last example for rational functions (click here):

                                              Find the domain of \f(x)=\frac{x+1}{x-1}

1. Set the denominator equal to 0

                                                                    x-1=0

2. Solve for x

                                                                   x-1=0

                                                                     x=1

                                              Domain:x<1 or x>1,or (-∞,1)∪(1,∞)

Finding the domains of functions isn’t too hard. As long as you follow the steps and practice finding the domain multiple times, you will be great at this. For more help or practice on this topic visit Symbolab’s Practice.

Until next time,

Leah