A function is a relation which states that for every possible x there is one and only one output for y. 

A relation is a function if it passes a vertical line test -- meaning that no vertical lines intersect the graph at more than one point. 

function	not a function

The notation f(x) means that we have a function named f and the variable in that function is x. (It does NOT mean "f times x.")

The domain of the function describes values of x that can be put into the function. The range of the function describes the values of the output. 

Evaluating a Function 

Evaluating a function means to substitute a variable with its given number or expression.

Example 

Given f(x) = 2x + 4, evaluate f(5).

This question asks you to substitute 5 for x and simplify.

f(5) = 2(5) + 4 = 14 

Adding and Subtracting Functions 

To add functions, add their outputs; to subtract functions, subtract their outputs. You can perform these operations on the functions themselves; just remember to simplify the resulting expressions.

Example 

Given f(x) = x² and g(x) = 4x + 1, find (f + g)(6) and (f − g)(x).

The expression (f + g)(6) really means f(6) + g(6). There are two ways you can do this-- you can evaluate f(6) and g(6) individually and add the results, or you can find (f + g)(x) and evaluate it at 6.

For the first option, we have f(6) = 6² = 36 and g(6) = 4(6) + 1 = 25. This gives us f(6) + g(6) = 36 + 25 = 61.

Using the other method, we have (f + g)(x) = (x²) + (4x + 1) = x² + 4x + 1. Evaluating, (f + g)(6) = 6² + 4(6) + 1 = 61.

The expression (f − g)(x) means f(x) − g(x). You find this function by subtracting the individual functions. Remember to use the distributive property when subtracting expressions.

(f − g)(x) = f(x) − g(x) = (x²) − (4x + 1) = x² − 4x − 1

Multiplying a Function by a Constant 

To multiply a function by a constant, multiply the output by that constant.  Remember to use the distributive property.

Example 

Given f(x) = 4x − 1, find 2f(x). 

 f(x) = 4x − 1 

 2f(x) = 2(4x − 1) = 8x − 2 

Multiplying a Function by a Function 

To multiply a function by another function, multiply the outputs. 

Example

Given f(x) = 2x and g(x) = x + 1, find (fg)(x) and (fg)(3).

The expression (fg)(x) means f(x)g(x). Multiply the expressions, using the distributive property when necessary.

(fg)(x) = f(x)g(x) = (2x) × (x + 1) = 2x(x + 1) = 2x² + 2x 

As with the other operations, you have the option of evaluating (fg)(3) by either multiplying f(3) and g(3) individually, or by evaluating (fg)(x) at 3.

f(3) = 2 × 3 = 6,  g(3) = 3 + 1 = 4,  (fg)(3) = 6 × 4 = 24

(fg)(x) = 2x² + 2x,  (fg)(3) = 2(3)² + 2(3) = 18 + 6 = 24

Practice

Given that f(x) = 3x² and g(x) = −2x + 7, find the following:

1.  f(8)

2.  f(2) + g(5)

3.  (f − g)(x)

4.  (f + 2g)(4)

5.  f(x)g(x)

6.  [g(x)]²

Answers

1.  192

2.  9

3.  3x² + 2x − 7

4.  46

5.  −6x³ + 21x²

6.  4x² − 28x + 49