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Math Lesson 16.4.4 - Function Composition with Itself

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Welcome to our Math lesson on Function Composition with Itself, this is the fourth lesson of our suite of math lessons covering the topic of Composite Functions, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Function Composition with Itself

It is possible to compose a function with itself. Suppose f(x) is a function, then the composition of function f with itself will be (f ∘ f) (x) = f [f(x)].

For example, if f(x) = 3x2, then

f ∘ f(x) = f[f(x)]
= 3 ⋅ (3x2)2
= 3 ∙ 9x4
= 27x4

Example 3

The functions f(x) = 3x + 5 and g(x) = 2x3 are given. Find:

  1. f ◦ g (1)
  2. g ◦ f (1)
  3. f ◦ f (2)
  4. g ◦ g (2)

Solution 3

First, we find the general form of the given composite function; then we substitute the value indicated by the question. Thus,

  1. f ∘ g(x) = f[g(x)]
    = 3 ⋅ (2x3 ) + 5
    = 6x3 + 5
    Therefore, for x = 1 we obtain:
    f ∘ g(1) = 6 ⋅ 13 + 5
    = 6 + 5
    = 11
  2. g ∘ f(x) = g[f(x)]
    = 2 ⋅ (3x + 5)3
    = 2 ⋅ [(3x)3 + 3 ⋅ (3x)2 ⋅ 5 + 3 ⋅ (3x) ⋅ 52 + 53]
    = 2 ∙ 27x3 + 2 ∙ 3 ⋅ 9x2 ∙ 5 + 2 ∙ 3 ∙ 3x ∙ 25 + 2 ∙ 125
    = 54x3 + 270x2 + 450x + 250
    Therefore, for x = 1 we obtain:
    g ∘ f(1) = 54 ∙ 13 + 270 ∙ 12 + 450 ∙ 1 + 250
    = 54 + 270 + 450 + 250
    = 1024
  3. We have to find f ◦ f (x) first. Thus,
    f ∘ f(x) = f[f(x)]
    = 3 ∙ (3x + 5) + 5
    = 9x + 15 + 5
    = 9x + 20
    Therefore, for x = 2 we obtain:
    f ∘ f(2) = 9 ∙ 2 + 20
    = 18 + 20
    = 38
  4. We have to find g ◦ g (x) first. Thus,
    g ∘ g(x) = g[g(x)]
    = 2 ∙ (2x3)3
    = 2 ⋅ (8x9)
    = 16x9
    Therefore, for x = 2 we obtain:
    g ∘ g(2) = 16 ∙ 29
    = 16 ∙ 512
    = 8192

We can also find the value of a composite function by considering the two original functions one by one. Thus, if we want to find f ◦ g (x) for x = a, then we find first g(a) = b, then we calculate f(b) (a and b are numbers). Let's explain this point through an example.

Example 4

Find f ◦ g (5) and g ◦ f (4) if f(x) = 4x2 and g(x) = 2x + 3.

Solution 4

In f ◦ g (5), we calculate g(5) first. Thus,

g(5) = 2 ∙ 5 + 3
= 10 + 3
= 13

Therefore, now we have to calculate f(13). We have

f(13) = 4 ∙ 132
= 4 ∙ 169
= 676

Hence, f ◦ g (5) = 676. Let's prove this result using the other method. We have

f ∘ g(x) = f[g(x)]
= 4 ⋅ (2x + 3)2
= 4 ∙ [(2x)2 + 2 ⋅ 2x ∙ 3 + 32]
= 4 ∙ (4x2 + 12x + 9)
= 16x2 + 48x + 36

Thus,

f ∘ g(5) = 16 ∙ 52 + 48 ∙ 5 + 36
= 16 ⋅ 25 + 48 ∙ 5 + 36
= 400 + 240 + 36
= 676

Hence, we obtain the same result but by using the one-by-one method it is possible to avoid dealing with big numbers.

You have reached the end of Math lesson 16.4.4 Function Composition with Itself. There are 9 lessons in this physics tutorial covering Composite Functions, you can access all the lessons from this tutorial below.

More Composite Functions Lessons and Learning Resources

Functions Learning Material
Tutorial IDMath Tutorial TitleTutorialVideo
Tutorial
Revision
Notes
Revision
Questions
16.4Composite Functions
Lesson IDMath Lesson TitleLessonVideo
Lesson
16.4.1The Meaning of Composite Function
16.4.2Properties of Composite Functions
16.4.3Evaluating Composite Functions
16.4.4Function Composition with Itself
16.4.5Showing a Composite Function Schematically
16.4.6Evaluating Composite Functions from a Graph
16.4.7Evaluating Composite Functions from a Table
16.4.8Domain of a Composite Function
16.4.9Range of a Composite Function

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