Function Notation

Until this point, all graphs have come from an equation that contains an x and a y variable. These variables indicate the input and output values and correspond to the x and y-axes. Now, functions will be modeled using a new notation.

For functions, "y =" and "f(x) =" (pronounced "f-of-x") mean the exact same thing. However, "f( x)" allows for more flexibility and more information. Instead of saying "y = 2x + 3; solve for y when x = -1", the new notation says "f(x) = 2x + 3; find f(-1)". In either case, the same thing needs to be done: substitute -1 for x, multiply by 2, and then add 3, simplifying to get an output value of 1. Using the first notation, the final statement becomes; when x = -1 then y = 1. In the second notation, this would be written f(-1) = 1.

Function notation allows greater flexibility than using just "y" for every formula. Graphing calculators will list different functions as y₁, y₂, etc. Textbooks use names like f(x), g(x), h(t), s(t), etc. With this notation, more than one function at a time can be used without confusing or mixing up the formulas.

A linear function can be written in the form where represents the output values of the equation or the y-values of the graph. The terms y and f(x) are often used interchangeably. For instance, the expression clearly shows that x is the independent variable. The values for the dependent variable, y, are calculated when values for x are substituted into the function.

Function notation is a very compact way of providing information. For example in the function f(3) = 6, the value inside the parentheses is the x value and the value the function equals is the dependent or y value. These two values yield an ordered pair that can be used for graphing the points on the line represented by the function.

Therefore, f(3) = 6 states when x = 3, then y = 6 or that the point (3, 6) is on the line. Review the table of values for both functions below:

Since both tables are the same, then f(x) = 2x is equivalent to y = 2x.

Evaluating a function is the process of substituting a value for the independent variable and simplifying the resulting numerical expression.

Example 1

Consider the linear function f(x) = 3x - 4. Find f(2).

Step 1. Substitute the given value of 2 for x.

f (2) = 3(2) - 4

Step 2. Follow the Order of Operations to simplify.

3(2) - 4

6 - 4

2

f (2) = 2

Evaluate f(2x-1) = 3x - 4.

Step 1. Substitute the contents of the parentheses for x.

f (2x-1) = 3(2x-1) - 4

Step 2. Use the Distributive Property.

3?2x + 3?(-1) - 4

Step 3. Simplify.

6x - 3 - 4

6x - 7

f (2x-1) = 6x - 7