How to Solve Composition of Functions? (+FREE Worksheet!)

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.

How to Solve Composition of Functions? (+FREE Worksheet!)

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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: \(\color{blue}{(f \ o \ g)(x)=f(g(x))}\)

Composition of Functions – 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 \(2: (f \ o \ g)(2)= f(g(2))\) \(=x-2=2-2=0\)

Composition of Functions – 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 \(g(x)\) by \(x+8\). Then: \(g(x+8)=(x+8)-2=x+8-2=x+6\)
Substitute \(x\) with \(4: (g \ o \ f)(4)= g(f(4))\) \(=x+6=4+6=10\)

Composition of Functions – Example 3:

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

Solution:

\((f \ o \ g)(x)=f(g(x)) \)
Then: \((f \ o \ g)(x)=f(g(x))=f(4x)\), now substitute \(x\) in \(f(x)\) by \(4x\). Then:

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

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

Exercises for Solving Composition of Functions

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

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

Download Composition of Functions Worksheet

Answers

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

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