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Composite Function

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Absolute Valued Functions

Quiz 

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Composite Functions

Functions can be combined to give a composite function (sometimes called a function of a function).

Example 1

Two functions are given f(x) = x + 1 and g(x) = 3x. Illustrate the composite function derived by operating with f first on x and then with g on the result.

Starting with x = 1, f(1) = 2. Now operating with g we have g[f(1))] = g(2) = 6.

Note that this is written as gf but f is the first function to operate, followed by g. (It does NOT mean g x f.

Instead of calculating the results one by one, we can produce a formula for the composite function gf.

So gf: x 3x + 3. gf(x) is not the same as fg(x). The order of a composite function is important.

Example 2

A function f is defined by f:x x² + 5x - 5 for x>0. Find the value of x which is unchanged by the mapping.

Since the image of x is x,

                 f(x) = x

i.e.   x² + 5x - 5 = x

        x² + 4x - 5 = 0

     (x + 5)(x - 1) = 0

                      x = -1 or 5

Since -5 is not in the domain, the value is 1.

Example 3

A function f is defined by f:x2x/(x - 1), x1.

(a) Obtain the expression for f² and f³.

(b) State the values for which the functions f² and f³ are not defined.

 

 

 

 

b) f² is not defined when x = 1 or -1

    f³ is not defined when x = 1 or -1 or 1/3.