Chapter Contents

• Functions and Graphs
• 1. Introduction to Functions
• 2. Functions from Verbal Statements
• 3. Rectangular Coordinates
• 4. The Graph of a Function
• Domain and Range of a Function
• Comparison calculator BMI - BAI
• 5. Graphing Using a Computer Algebra System
• Online graphing calculator (1): Plot your own graph (JSXGraph)
• Online graphing calculator (2): Plot your own graph (SVG)
• 6. Graphs of Functions Defined by Tables of Data
• 7. Continuous and Discontinuous Functions
• 8. Split Functions
• 9. Even and Odd Functions
• Functions and Graphs Problem Solver

• Comments, Questions?

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Domain and Range of a Function

Definitions of Domain and Range

Domain

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

In plain English, this definition means:

The domain of a function is the set of all possible x-values which will make the function "work" and will output real y-values.

When finding the domain, remember:

• The denominator (bottom) of a fraction cannot be zero
• The values under a square root sign must be positive

Example 1a - Domain

The function `y = sqrt(x+4)`  has the following graph.

The domain of the function is x ≥ −4, since x cannot take values less than −4. To see why, try some values in your calculator, some less than −4 (like −5 or −10) and some more than −4 (like −2 or 8) . The only ones that "work" and give us an answer are the ones greater than or equal to −4. This will make the values under the square root positive.

Notes:

1. The enclosed (colored-in) circle on the point (-4, 0). This indicates that the domain "starts" at this point.
2. We saw how to draw similar graphs in section 4, Graph of a Function. For a more advanced discussion, see also How to draw y^2 = x − 2.

Range

The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually) of a function, after we have substituted the domain values.

In plain English, the definition means:

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

When finding the range, remember:

• Substitute different x-values into the expression for y to see what is happening. (Is y always positive? Always negative? Or maybe not equal to certain values?)
• Make sure you look for minimum and maximum values of y.
• Draw a sketch! In math, it's very true that a picture is worth a thousand words.

Example 1b - Range

Let's return to the example above, `y = sqrt(x + 4)`.

We notice that there are only positive y-values, and zero. No matter what value of x we try, we will always get a zero or positive value of y. We say the range for this function is y ≥ 0.

The squiggle at the top of the arrow in the graph indicates the range goes on forever, beyond what is shown on the graph.

Example 2

Consider the following curve, y = sin x.

The domain of the function y = sin x is "all values of x", since there are no restrictions on the values for x. (Put any number into your "sin" function in your calculator. Any number should work, and will give you a value between −1 and 1.

From the calculator experiment, and from observing the curve, we can see the range of the function is y betweeen −1 and 1. We could write this domain as −1 ≤ y ≤ 1.

Where did this graph come from? We learn about sin and cos graphs later in Graphs of sin x and cos x

Note 1: Because we are assuming that only real numbers are to be used in the domain and range of a function, values that lead to division by zero or to imaginary numbers (which arise from finding the square root of a negative number) are not included. The Complex Numbers chapter explains more about imaginary numbers.

Note 2: When doing square root examples, many people ask, "Don't we get a positive and a negative answer when we find a square root?" A square root has at most one value, not two. See this discussion: Square Root 16 - how many answers?

Note 3: We are talking about the domain and range of functions, which have at most one y-value for each x-value.

Summary - How to find the domain

In general, we determine the domain of each function by looking for those values of the independent variable (usually x) which we are allowed to use. (We have to avoid 0 on the bottom of a fraction, or negative values under the square root sign).

Summary - How to find the range

The range of each function is found through deciding what the resulting y-values are after we have substituted in our possible x-values.

Exercise 1

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

Answer

Exercise 2

Find the domain and range for the function

`f(t)=1/(t+2)`

Answer

Exercise 3

Find the domain and range for the function

`g(s)=sqrt(3-s)`

Answer

Exercise 4

Find the domain and range for the function defined as

f(x) = x² + 4 for x > 2

Answer

Exercise 5

We fire a ball up in the air and find the height h, in metres, as a function of time t, in seconds, is given by

h = 20t − 4.9t²

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

Answer

Functions defined by coordinates

Sometimes we don't have continuous functions. How do we find the domain and range? Let's look at an example.

Exercise 6

Find the domain and range of the function defined by the coordinates:

{(−4, 1), (−2, 2.5), (2, −1), (3, 2)}

Answer