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Domain and Range of a Function

Example 1

Given the function:

a. Find the domain.

b. Find the range.

Solution

a. To find the domain, ask yourself, “What is x allowed to be?”. The square root of a number results in a real number only if the radicand is greater than or equal to zero.

Thus, x - 3 must be greater than or equal to 0.

Add 3 to both sides.

x - 3 ≥ 0

x ≥ 3

So, we can write the domain of as x 3.

Using interval notation, this is written [3, +).

b. To determine the range, find all possible y-values for the x-values in the domain.

Taking a square root always results in a nonnegative number. Thus, the smallest can be is 0 (that happens when x = 3).

When x is larger than 3, is a positive number.

So, the range of y = is y 0.

Note:

 

The symbol + means “positive infinity.” Since infinity is greater than any real number, we must use a parenthesis when using in interval notation. For example:

x > 5 in interval notation is (5, +)

x 7 in interval notation is (-, 7]

“All real numbers” can be expressed in interval notation as (-, + ).

 

Example 2

Given the function:

a. Find the domain.

b. Find the range.

Solution

a. To find the domain, ask yourself, “What is x allowed to be?”.

We can divide by any number except 0.

Thus, we must determine when the denominator is 0.

Add x to both sides.

5 - x = 0

5 = x

So, the domain is all real numbers except 5. We write this as “all real numbers where x 5”. We can also write the domain as (-, 5) (5, + ). The symbol, , means union. The union of two sets is all the elements in both sets combined.

Notice that the graph gets closer and closer to the vertical line x = 5, but never touches it. We call the line x = 5 an asymptote. An asymptote is a line that the function approaches but never reaches.

b. To find the range, note that no matter how large x gets in the value of this fraction will never be 0. We can also see this from the graph. The line y = 0 is an asymptote.

So, the range of y = is all real numbers except 0.

We can write the range as “all real numbers where y 0” or as (-, 0) (0, +).

Note:

We can write

 

 
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