Informative line

Composite Functions

Learn how to do composite functions & fog x & fx functions, Practice to finding the composition of functions when two functions are given in form of tables.

Evaluating Composite Functions 

To find \(fog(x)\) at a particular value of \(x\), say \(x=a\), put \(x=a\), evaluate \(g(a)\) and then put this value in the expression of \(f\).

Illustration Questions

Let \(f(x)=\sqrt x\) and \(g(x)=\sqrt {3-x}\), then the value of \(fog(-1)\) is 

A \(5\)

B \(\sqrt 7\)

C \(-1\)

D \(\sqrt 2\)

×

\(fog(-1)=f(g(-1))\)

\(g(-1)=\sqrt {3-(-1)}=\sqrt 4=2\)

\(f(g(-1))=f(2)=\sqrt 2\)

Let \(f(x)=\sqrt x\) and \(g(x)=\sqrt {3-x}\), then the value of \(fog(-1)\) is 

A

\(5\)

.

B

\(\sqrt 7\)

C

\(-1\)

D

\(\sqrt 2\)

Option D is Correct

Evaluating Composite Function when Two Function are given in Form of Tables

  • If values of two function \(f\) and \(g\) are given in tabular form for some values of \(x\), we can find the value of composite of these two function at a particular value of \(x\).

e.g.

Consider the value of \(fog(8)=f(g(8))\)

\(\Rightarrow f(7)=6\)

Illustration Questions

Consider the table. The first row indicates values of \(x\) and second and third row gives values of \(f(x)\) and \( g(x)\) for corresponding \(x\)above it in the table. What is the value of \(f(g(3))\) ?

A 2

B 6

C 5

D 1

×

From the table,

 \(g(3)=1\) (Circled value)

image

\(\Rightarrow f(g(3))=f(1)=1\) (Value in the square)

image

\(\Rightarrow f(g(3))=1\)

image

Consider the table. The first row indicates values of \(x\) and second and third row gives values of \(f(x)\) and \( g(x)\) for corresponding \(x\)above it in the table. What is the value of \(f(g(3))\) ?

image
A

2

.

B

6

C

5

D

1

Option D is Correct

Evaluation of Composite Function when the Graphs of two Functions are given

  • Given the graph of two functions \(f\) and \(g\) if we need to find out the value of composite function of \(f\) and \(g\) at some value of \(x\) say \(f(g(\alpha))\), we find \(g(\alpha)\) first by noting the height of graph of \(g\) above the point \((\alpha, 0)\) and then the height of graph of \(f\) above the point \((g(\alpha), 0)\).
  • e.g.

Illustration Questions

Consider the graph of two function \('f'\) and \('g'\). From the above graph the value of \(fog(6)\) is 

A 0

B 7

C –8

D 9

×

\(fog(6)=f(g(6))\)

\(g(6)\) is the height of graph at \((6, 0)\) which is 3.

\(\therefore \;g(6)=3\)

Now \(fog(6)=f(3)=0\rightarrow\) graph of \('f'\) is at \(x\) axis when \(x=3\).

Consider the graph of two function \('f'\) and \('g'\). From the above graph the value of \(fog(6)\) is 

image
A

0

.

B

7

C

–8

D

9

Option A is Correct

Composite Function of More than Two Functions

  • The symbol \(f\,o\,g\,\,o\,h=f(g(h(x)))\).
  • We evaluate \(h(x)\) first, then put the value in the rule of \(g\) to get \(g(h(x))\) and then put this value in the rule of \(f\) to get \(f[g(h(x))]\)

Illustration Questions

Consider \(f(x)=\dfrac {2x}{x+1}\), \(g(x)=x^4\) and \(h(x)=x+2\) then  \(f\,o\,g\,\,o\,h\;(x)\) is given by the expression

A \(\dfrac {2x}{x^4+1}\)

B \(\dfrac {x^4}{2x+1}\)

C \(\dfrac {2(x+2)^4}{(x+2)^4+1}\)

D \(x^5\)

×

\(f\;o\;g\;o\;h(x)=f(g(h(x))\)

\(g(h(x))=g(x+2)\)

\(g(x+2)=(x+2)^4\)

\(f(g(h(x))=f((x+2)^4)=\dfrac {2(x+2)^4}{(x+2)^4+1}\)

Consider \(f(x)=\dfrac {2x}{x+1}\), \(g(x)=x^4\) and \(h(x)=x+2\) then  \(f\,o\,g\,\,o\,h\;(x)\) is given by the expression

A

\(\dfrac {2x}{x^4+1}\)

.

B

\(\dfrac {x^4}{2x+1}\)

C

\(\dfrac {2(x+2)^4}{(x+2)^4+1}\)

D

\(x^5\)

Option C is Correct

Finding the Component Functions from the Composite Function

  • Sometimes it is very useful if we know how to find the component function given a complicated composite function. In other words we should be able to find \(f\) and \(g\) by looking at \(fog \)

Illustration Questions

Let \(h(x)=\dfrac {sin\,x}{2+sin\,x}\) be a composite function. If it is expressed in the form \(fog(x)\) then

A \(f(x)=\dfrac {x}{1+x}\) and \(g(x)=sin\,x\) 

B \(f(x)=\dfrac {1+x^2}{x}\) and \(g(x)=cos\,x\)

C \(f(x)=cos\,x\) and \(g(x)=x^3+3\)

D \(f(x)=\dfrac {1}{x}\) and \(g(x)=cos\,x\)

×

If \(f(g(x))=\dfrac {sin\,x}{1+sin\,x}=\dfrac {t}{1+t}\), where \(t=sin\,x\)  

\(sin\,x\) is occurring more than once in the expression.

\(\therefore \;\;g(x)=t\)

\(\therefore \;\;f(t)=\dfrac {t}{1+t}\Rightarrow f(x)=\dfrac {x}{1+x}\) and \(g(x)=sin\,x\)

Note: Most of these problem be a composite function will be solved by inspection .

Let \(h(x)=\dfrac {sin\,x}{2+sin\,x}\) be a composite function. If it is expressed in the form \(fog(x)\) then

A

\(f(x)=\dfrac {x}{1+x}\) and \(g(x)=sin\,x\) 

.

B

\(f(x)=\dfrac {1+x^2}{x}\) and \(g(x)=cos\,x\)

C

\(f(x)=cos\,x\) and \(g(x)=x^3+3\)

D

\(f(x)=\dfrac {1}{x}\) and \(g(x)=cos\,x\)

Option A is Correct

Breakup of a Function which is a Composite of More than two Functions

  • If a composite function expression of more than two functions is given then the component functions can be found by inspection.

e.g. 

Let \(f(g(h(x))=sin(cos\,x^2)\), then we observe that the rightmost expression is \(x^2\) so \(h(x)=x^2\), to the left of it is \(cos\) so \(g(x)=cos\,x\) and leftmost function is \(sin\). So \(f(x)=sin\,x\)

 

Illustration Questions

If \(f\,o\,g\,o\,h(x)=sin^4\sqrt x\), then which of the following is correct ?

A \(f(x)=x^4,\;g(x)=sin\,x, \;h(x)=\sqrt x\)

B \(f(x)=cos\,x,\;g(x)=x^2, \;h(x)=\dfrac {1}{x}\)

C \(f(x)=x+1,\;g(x)=tan\,x,\; h(x)=x^3\)

D \(f(x)=\sqrt [3] x,\;g(x)=cos\,x, \;h(x)=\dfrac {1}{x}\)

×

\(f\,o\,g\,o\,h(x)=sin^4\sqrt x\)

Note the three functions in sight, if we evaluate this expression what operation will be performed first, it will be taking square root. So inner most function \(h(x)=\sqrt x\)

 By the same  thought second operation is \(sin\) so \(g(x)=sin\,x\) and lastly we perform raising to power \(4\).

So  \(f(x)=x^4\)

If \(f\,o\,g\,o\,h(x)=sin^4\sqrt x\), then which of the following is correct ?

A

\(f(x)=x^4,\;g(x)=sin\,x, \;h(x)=\sqrt x\)

.

B

\(f(x)=cos\,x,\;g(x)=x^2, \;h(x)=\dfrac {1}{x}\)

C

\(f(x)=x+1,\;g(x)=tan\,x,\; h(x)=x^3\)

D

\(f(x)=\sqrt [3] x,\;g(x)=cos\,x, \;h(x)=\dfrac {1}{x}\)

Option A is Correct

Finding one Component of a Composite Function, given the composite function and its other component

  • Sometimes we are given the composite function expression of two functions \(f\) and \(g\) i.e. \(gof(x)\) and also the expression for \(f(x)\).To find \(g(x)\) in such case,put \(f(x)=t\) and express the \(gof(x)\) expression in terms of \(t\).

e.g. 

\(f(g(x))=5x+3\) and \(g(x)=2x+1\), then \(f(2x+1)=5x+3\)

Put \(2x+1=t\Rightarrow x=\dfrac {t-1}{2}\)

\(\therefore f(t)=5\dfrac {(t-1)}{2}+3=\dfrac {5t-5+6}{2}\)

\(\Rightarrow f(t)=\dfrac {5t+1}{2}\Rightarrow f(x)=\dfrac {5x+1}{2}\)

Illustration Questions

Let \(f(x)=x^2+2\) and \(gof(x)=3x^2+7\), then which of the following function will be \(g(x)\)?

A \(g(x)=3x+1\)

B \(g(x)=4x^2+7\)

C \(g(x)=sin\,x+2\)

D \(g(x)=\dfrac {1}{x}\)

×

\(g(f(x))=3x^2+7\)

\(\Rightarrow g(x^2+2)=3x^2+7\)

Put \(x^2+2=t\Rightarrow x^2=t-2\)

Now put  \(x^2\)  in terms of  \(t\) on R.H.S.

\(\Rightarrow g(t)=3(t-2)+7\)

\(\Rightarrow g(t)=3t-6+7 \;\Rightarrow g(t)=3t+1\)

\(\Rightarrow g(x)=3x+1\)

Let \(f(x)=x^2+2\) and \(gof(x)=3x^2+7\), then which of the following function will be \(g(x)\)?

A

\(g(x)=3x+1\)

.

B

\(g(x)=4x^2+7\)

C

\(g(x)=sin\,x+2\)

D

\(g(x)=\dfrac {1}{x}\)

Option A is Correct

Value of Composite Function of more than two Function at a Particular Value of x

  • Consider two function \(f(x)\) and \(g(x)\), we define composite function of \('f'\)  and \('g'\) as \(fog(x)=f(g(x))\) (This is different from \((fg)(x)\)).
  • First \('g'\) is applied to \(x\) and then \('f'\) rule is applied to \(g(x)\) .

  • The output of \('g'\) machine acts as an input for \('f'\) machine and final output is \(f(g(x))\).
  • Similarly we can define \(g(f(x))=gof(x)\) or \(fof(x)=f(f(x))\)

  • Similarly the composite of three function \(f, \,g, \,h\) is defined as  \(fogoh(x)=f(g(h(x))\)

 

Illustration Questions

If \(f(x)=2x+3\) , \(g(x)=5x+1\) and \(h(x)=x^2+2x+3\) find the value of \(fogoh(2)\).

A 72

B 115

C 95

D 47

×

\(fogoh(x)=f(g(h(x))\)

In this case,

\(f(x)=2x+3\),  \(g(x)=5x+1\)

\(h(x)=x^2+2x+3\)

\(\therefore \;fogoh(2)=f(g(h(2)))=f(g(4+4+3))\)

\(=f(g(11))=f(5×11+1)=f(56)=2×56+3=115\)

\(\therefore \;fogoh(2)=115\)

If \(f(x)=2x+3\) , \(g(x)=5x+1\) and \(h(x)=x^2+2x+3\) find the value of \(fogoh(2)\).

A

72

.

B

115

C

95

D

47

Option B is Correct

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