Composition And Inverse Functions

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Composition of Functions

In mathematics, it is often the case that the result of one function is evaluated by applying a second function. For example, consider the functions defined by f(x)=x2 and g(x)=2x+5. First, g is evaluated where x=−1 and then the result is squared using the second function, f.

This sequential calculation results in 9. We can streamline this process by creating a new function defined by f(g(x)), which is explicitly obtained by substituting g(x) into f(x).

f(g(x))=f(2x+5)=(2x+5)2=4x2+20x+25

Therefore, f(g(x))=4x2+20x+25 and we can verify that when x=−1 the result is 9.

f(g(−1))=4(−1)2+20(−1)+25=4−20+25=9     

The calculation above describes composition of functionsApplying a function to the results of another function., which is indicated using the composition operatorThe open dot used to indicate the function composition (f○g)(x)=f(g(x)). (○). If given functions f and g,

(f○g)(x)=f(g(x))       Composition of Functions

The notation f○g is read, “f composed with g.” This operation is only defined for values, x, in the domain of g such that g(x) is in the domain of f.

Example 1

Given f(x)=x2−x+3 and g(x)=2x−1 calculate:

  1. (f○g)(x).
  2. (g○f)(x).

Solution:

  1. Substitute g into f.

    (f○g)(x)=f(g(x))=f(2x−1)=(2x−1)2−(2x−1)+3=4x2−4x+1−2x+1+3=4x2−6x+5

  2. Substitute f into g.

    (g○f)(x)=g(f(x))=g(x2−x+3)=2(x2−x+3)−1=2x2−2x+6−1=2x2−2x+5

Answer:

  1. (f○g)(x)=4x2−6x+5
  2. (g○f)(x)=2x2−2x+5

The previous example shows that composition of functions is not necessarily commutative.

Example 2

Given f(x)=x3+1 and g(x)=3x−13 find (f○g)(4).

Solution:

Begin by finding (f○g)(x).

(f○g)(x)=f(g(x))=f(3x−13)=(3x−13)3+1=3x−1+1=3x

Next, substitute 4 in for x.

(f○g)(x)=3x(f○g)(4)=3(4)=12

Answer: (f○g)(4)=12

Functions can be composed with themselves.

Example 3

Given f(x)=x2−2 find (f○f)(x).

Solution:

(f○f)(x)=f(f(x))=f(x2−2)=(x2−2)2−2=x4−4x2+4−2=x4−4x2+2

Answer: (f○f)(x)=x4−4x2+2

Try this! Given f(x)=2x+3 and g(x)=x−1 find (f○g)(5).

Answer: 7

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