Composition of Functions

The “composition of functions” is the process of combining two or more functions in a way where the output of one function is the input for the next function.

Step by step guide to solve Composition of Functions

• The term “composition of functions” is simply the combination of two or more functions where the output from one function becomes the input for the next function.
• The notation used for composition is: $$\color{blue}{(f \ o \ g)(x)=f(g(x))}$$

Example 1:

Using $$f(x)=x-2$$ and $$g(x)=x$$, find: $$f(g(2))$$

Solution:

$$(f \ o \ g)(x)=f(g(x))$$
Then: $$(f \ o \ g)(x)=f(g(x))=f(x)=x-2$$
Substitute $$x$$ with $$2: (f \ o \ g)(2)=2-2=0$$

Example 2:

Using $$f(x)=x+8$$ and $$g(x)=x-2$$, find: $$g(f(4))$$

Solution:

$$(f \ o \ g)(x)=f(g(x))$$
Then: $$(g \ o \ f)(x)=g(f(x))=g(x+8)$$, now substitute $$x$$ in $$f(x)$$ by $$x+8$$. Then: $$g(x+8)=(x+8)-2=x+8-2=x+6$$
Substitute $$x$$ with $$4: (g \ o \ f)(4)=g(f(x))=4+6=10$$

Example 3:

Using $$f(x)=x+2$$ and $$g(x)=4x$$, find: $$f(g(1))$$

Solution:

$$(f \ o \ g)(x)=f(g(x))$$
Then: $$(f \ o \ g)(x)=f(g(x))=f(4x)=4x+2$$
Substitute $$x$$ with $$1: (f \ o \ g)(1)=4+2=6$$

Example 4:

Using $$f(x)=5x+4$$ and $$g(x)=x-3$$, find: $$g(f(3))$$

Solution:

$$(f \ o \ g)(x)=f(g(x))$$
Then: $$(f \ o \ g)(x)=g(f(x))=g(5x+4)$$, now substitute $$x$$ in $$g(x)$$ by $$5x+4$$. Then: $$g(5x+4)=(5x+4)-3=5x+4-3=5x+1$$
Substitute $$x$$ with $$3: (g o f)(x)=g(f(x))=5x+1=5(3)+1=15=1=16$$

Exercises

Using $$f(x) = 5x + 4$$ and $$g(x) = x – 3$$, find:

• $$\color{blue}{f(g(6))}$$
• $$\color{blue}{f(f(8))}$$
• $$\color{blue}{g(f(– 7))}$$
• $$\color{blue}{g(f(x))}$$

• $$\color{blue}{19}$$
• $$\color{blue}{224}$$
• $$\color{blue}{-34}$$
• $$\color{blue}{5x+1}$$

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