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Wednesday 29th of June
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 Depdendent Variable

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 Dependent Variable

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Composition of Functions

Another way to combine two functions is called the composition of functions. This is where the output of one function becomes the input of a second function.

Definitionâ€” Composition of Functions

Let f(x) and g(x) represent two functions. The composition of f and g, written (f g)(x), is defined as

(f g)(x) = f[g(x)]

Here, g(x) must be in the domain of f(x). If it is not, then f[g(x)] will be undefined.

Note:

In the composition (f g)(x), the output from g(x) is the input for f(x).

Example 1

Given f(x) = 3x - 5 and g(x) = x2 - 7, find:

a. (f g)(x)

b. (g f)(x)

Solution

 a. Use g(x) as the input for f(x). Replace g(x) with x2 - 7. In f(x), replace x with x2 - 7. Remove parentheses. Subtract. So, (f ○ g) = 3x2 - 26. (f ○ g)(x) = f[g(x)]= f[x2 - 7] = 3(x2 - 7) - 5 = 3x2 - 21 - 5 = 3x2 - 26 b. Use f(x) as the input for g(x). Replace f(x) with 3x - 5. In g(x), replace x with 3x - 5. Square the binomial. Combine like terms. (g ○ f)(x) = g[f(x)] = g[3x - 5] = (3x - 5)2 - 7 = 9x2 - 15x - 15x + 25 - 7 = 9x2 - 30x + 18

So, (g ○ f)(x) = 9x2 - 30x + 18.

Notes:

In (f g)(x) = f[g(x)], we substitute the expression for g(x) into f(x).

In (g f)(x) = g[f(x)], we substitute the expression for f(x) into g(x).

In this example:

(f g)(x) = 3x2 - 19

(g f)(x) = 9x2 - 30x + 18.

In most cases, (f g)(x) and (g f)(x) do NOT yield the same result.