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 $$f$$.

#### 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 $$fog(8)=f(g(8))$$

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

#### 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)

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

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

### 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

Option D 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 $$fog$$

#### 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

# 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  $$fogoh(x)=f(g(h(x))$$

#### 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

# 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.

#### 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

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 $$f[g(h(x))]$$

#### 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

# 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 $$f(x)=sin\,x$$

#### 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 $$2x+1=t\Rightarrow x=\dfrac {t-1}{2}$$

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

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

#### 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