### 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) = x2 + 2, then -2h(x) = - 2(x2 +2) = - 2x2 - 4. (3x)h(x) = (3x)(x2 +2) = 3x3 + 6x.

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) = 2x2 + 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) = x2 - 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 (x2 -8) = 2(x2 -8) - 2 = 2x2 - 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 = 4x2 - 8x - 4.

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