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 Depdendent Variable

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 Dependent Variable

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# Functions and Graphs

## Definition of a Function

We often work with two quantities that are related.

For example, letâ€™s say you put gas in your car. The total cost of the gas is related to the number of gallons of gas you buy.

Suppose gas costs \$1.50 per gallon. Then, the relationship between the cost in dollars, y, and the number of gallons, x, is given by: y = 1.50x

The equation y = 1.50x is an example of a function.

Definition â€” Function

A function is a rule that assigns to each number in one set exactly one number in another set.

The function y = 1.50x is a rule that says to start with a number, represented by x, and to multiply that number by 1.50. The result is the number represented by y. The ordered pair, (x, y) is said to satisfy the function.

For example, the ordered pair (2, 3) satisfies the function y = 1.50x. This is because replacing x with 2 and y with 3 results in a true statement:

Is 3 = 1.50 Â· 2 ?

Is 3 = 3 ? Yes

Example 1

In the function y = 1.50x, the variable y represents the cost in dollars and x represents the number of gallons of gas purchased. Find the cost of 6 gallons of gas.

Solution

In the function, y = 1.50x, substitute 6 for x and simplify.

 Let x = 6Multiply y = 1.50 (6)y = 9.00
So, the cost of 6 gallons of gas is \$9.00.

The ordered pair (6, 9) satisfies the function y = 1.50x.

The two sets related by a function have special names. The first set is called the domain of the function. A number in this set is called an input. The input is usually represented by the letter x.

The second set is called the range of the function. A number in this set is called an output. The output is often represented by the letter y.

Note:

You can think of a function as a machine that assigns to each input number, x, exactly one output number, y.

Here are three more examples of functions:

 Absolute Value function Quadratic  function Square Root function y = - x2 - 6 y = 4|x| - 8

## Function Notation

Functions can also be written in function notation, a special notation that uses letters such as f, g, and h.

For example, when x is the input, the output may be written as f(x). This is read, â€œf of xâ€. This notation shows that each output value is related to each input value by the function, f.

For example,

y = - x2 - 6 is equivalent to f(x) = - x2 - 6

y = 4|x| - 8 is equivalent to g(x) = 4|x| - 8

is equivalent to

If we are given a value for x, we can find the corresponding value of the function, f(x).

Note:

In the past, you have used parentheses to indicate multiplication. Thus, f(x) could also be interpreted as the variable f multiplied by the variable x. The only way to tell if f(x) means function or multiplication is to look at the context in which it is used.