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Multiplication and Composition of Functions

Multiplication and Composition of Functions

Multiplication and Composition of Functions

Multiplication and Composition of Functions

Multiplication and Composition of 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.