# Operations with Functions

### Multiplication and Composition of Functions

#### Scalar Multiplication

To multiply a function by a scalar, multiply each output by that scalar. For example, if f (x) = 4x - 1 , then f (x) = (4x - 1) = 2x - . If g(x) = x - 2 , then 3g(x) = 3(x - 2) = 3x - 6 . If h(x) = x 2 + 2 , then -2h(x) = - 2(x 2 +2) = - 2x 2 - 4 . (3x)h(x) = (3x)(x 2 +2) = 3x 3 + 6x .

Scalar Multiplication

The y -coordinate of each point on the graph of f (x) is the result of multiplying the y -coordinate of f by .

#### Multiplication of Functions

To multiply a function by another function, multiply their outputs. For example, if f (x) = 2x and g(x) = x + 1 , then fg(3) = f (3)×g(3) = 6×4 = 24 . fg(x) = 2x(x + 1) = 2x 2 + x .

#### Compound Functions

When we take f (g(x)) , we take g(x) as the input of the function f . In other words, we take x as the input of g and evaluate g(x) , and then we take this result as the input of f and evaluate f (g(x)) .

For example, if f (x) = 10x and g(x) = x + 1 , then to find f (g(4)) , we find g(4) = 4 + 1 + 5 , and then evaluate f (5) = 10(5) = 50 . Similarly, f (g(12)) = f (12 + 1) = f (13) = 10(13) = 130 . In general, f (g(x)) = f (x + 1) = 10(x + 1) = 10x + 10 .

Example: f (x) = 2x - 2 , g(x) = x 2 - 8 .

f (g(3)) = f (32 - 8) = f (1) = 0 .
f (g(- 4)) = f ((- 4)2 - 8) = f (8) = 2(8) - 2 = 14 .

In general, f (g(x)) = f (x 2 -8) = 2(x 2 -8) - 2 = 2x 2 - 18 .
g(f (3)) = g(2(3) - 2) = g(4) = 42 - 8 = 8 .
g(f (- 4)) = g(2(- 4) - 2) = g(- 10) = (- 10)2 - 8 = 92 .

In general, g(f (x)) = g(2x - 2) = (2x - 2)2 -8 = 4x 2 - 8x - 4 .

f (g(x)) is denoted f o g(x) and g(f (x)) is denoted g o f (x) . Note that it is not necessarily true that f o g(x) = g o f (x) , as shown in the above example.