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 Depdendent Variable

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 Dependent Variable

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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, â€œWhat is x allowed to be?â€. 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 â€œpositive infinity.â€ 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]

â€œAll real numbersâ€ 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, â€œWhat is x allowed to be?â€.

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 â€œall real numbers where x 5â€. 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 â€œall real numbers where y 0â€ or as (-, 0) (0, +).

Note:

We can write