2a. Domain and Range of a Function

Definitions of Domain and Range

Domain: The complete set of possible values of the independent variable in a function is called the domain of the function.

In plain English, the definition means:

The domain of a function is all the possible x values which will make the function "work" by outputting real values.

When finding the domain, remember:

denominator (bottom) of a fraction cannot be zero

the values under a square root must be positive

Range: The complete set of all possible resulting values of the dependent variable in a function is the range of the function.

In plain English, the definition means:

The range of a function is the possible y values of a function resulting when we substitute all the possible x-values.

When finding the range, remember:

substitute different x-values into the expression for y to see what is happening

make sure you look for minimum and maximum values of y

draw a sketch!

Because real numbers are used in the domain and range of a function, values that lead to division by zero or to imaginary numbers are not included.

Imaginary Numbers

An imaginary number is the result of finding the square root of a negative number. [Your calculator may give you an ERROR message if you try to do this, but it is still possible.]

Imaginary numbers have applications in electronics.

Example

√(-4) = 2j

The number 2j is an imaginary number.

More Domain and Range Examples

You can see more examples of domain and range in the section Inverse Trigonometric Functions.

The Complex Numbers chapter explains more about imaginary numbers.

Example 1

(a) Find the domain and range for the function
f(x) = x² + 2.

Answer

(b) Find the domain and range for the function

Answer

Example 2

Find the domain and range for the function

Answer

In general, we determine the domain of each function by looking for those values of the independent variable which cannot be used.

The range of each function is found through an inspection of the function.

Example 3

Find the domain and range for the function defined as

f(x) = x² + 4 for x > 2

Answer

Example 4

More Domain and Range Examples

In case you missed it earlier, you can see more examples of domain and range in the section Inverse Trigonometric Functions.

We are told that the height h, in metres, of a certain projectile as a function of time t, in seconds, is

h = 20t − 4.9t²

Find the domain and range for the function h(t).

Answer

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