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Tutorials that teach
Finding the Domain and Range of Functions

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Tutorial

- Domain and Range
- Domain and Range of Square Root Functions
- Domain and Range of Rational Functions

**Domain and Range**

Domain: the set of input values of a function or relation

Range: the set of output values of a function or relation

In a function, the domain is the set of all possible input values. In other words, it represents all values that the independent variable (usually x) is allowed to take on in order to return a value for the function. The range is the set of all possible output values; or values that the function will have, depending on all of the possible input values.

When a function has no restrictions on its domain, it is continuous from negative infinity to positive infinity along the x-axis. If the range also has no restrictions, then on one end, the graph tends towards negative infinity, and on the other end, it tends towards positive infinity. Below is the graph of polynomial that has no domain or range restrictions:

When talking about domain restrictions, the two common cases are when there is an expression underneath a square root (or other even root), and when there is a denominator with a variable in it (as is the case with rational functions). These represent domain restrictions because the expression under the even-root must be non-negative, and the denominator must not equal zero. We will use these ideas in finding domain restrictions below.

**Domain and Range of Square Root Functions**

Find the domain and range of

The find the domain, we need to identify restrictions to the domain. Since this is a square root function, we need to set the expression underneath the radical greater than or equal to zero, and solve the inequality for x:

This means that the function is defined for any x-value greater than or equal to negative one-half.

If the inequality has no solution (that is, no values of x for which the expression is less than zero), then the domain has no restrictions.

We can example the graph of this function to describe its range:

We see that as x gets larger and larger, the function heads towards positive infinity (even if rather slowly). On the other side of the graph, however, the function never falls below the x-axis. Therefore, the range is from zero to positive infinity. Since the function can have the exact value of zero, we can write this as [0, ∞)

**Domain and Range of Rational Functions**

Find the domain and range of

With rational functions, our domain restrictions are x-values that make the denominator equal to zero. So we need to set the denominator equal to zero and solve for x. This will give us x-values to exclude from the domain:

This means that the domain of the rational function is all x–values *except* x = –1, and x = 2

If the equation has no solution (that is, no values of x make the denominator equal zero) then the domain has no restrictions.

To find the range, we can examine the graph of this function, too:

We can see that the domain restrictions represent vertical asymptotes in this function. We also see a horizontal asymptote, which can be found by dividing the leading coefficients in the numerator and denominator of the function. So the line is y = 2. We may be tempted to say that 2 is excluded from the range, because as the function gets more and more negative, and more and more positive, the value of the function approaches, but never reaches 2. However, in between our vertical asymptotes, we see that function actually does at one point have a value of 2. We can say, then, that the range of this function is all real numbers, (–∞, ∞).