Composition of Functions MCQ Quiz - Objective Question with Answer for Composition of Functions - Download Free PDF

Explanation -

g(x) = x2 + x – 1

gof(x) = 4x2 – 10x + 5

g(f(x) = 4x2 – 10x + 5

f2(x) + f(x) – 1 = 4x2 – 10x + 5

Putting x = 5/4 and f(5/4) = t

t = -1/2 or f(5/4) = -1/2

Hence Option (2) is correct.

Explanation:

Substitute the value of to get the following expression:

(3)

will be the same as

will be the same as

Explanation:

Given:

fog(x) =cos 2 √x and go f(x) = |cos x|. 

If we take option (c),

⇒ f(x) = cos² x; then g(x) = √x 

⇒ fog(x) = cos 2 √x and go f(x) = 

= |cos²x|

Then f(x) = cos² x

If f(x) = cos² x then g(x) = √ x

∴ Option (a) is correct.

Explanation:

Given:

fog(x) =cos 2 √x and go f(x) = |cos x|. 

If we take option (c),

⇒ f(x) = cos² x; then g(x) = √x 

⇒ fog(x) = cos 2 √x and go f(x) = 

= |cos²x|

Then f(x) = cos² x

∴ Option (c) is correct.

Explanation:

Given:

f(x) = 9x - 8√x 

g(x) = f(x) - 1

Let x = t²

⇒ g(t² ) = 9t² – 8t – 1 = 0

⇒ t = 

⇒ x = 1 , 1/81

 g(x) = 0 has only one real root which is an integer.

∴ Option (b) is correct.

Concept:

If f :A → B and g : C → D. Then (fog) (x) will exist if and only if the co-domain of g = domain of f i.e D = A and (gof) (x) will exist if and only if co-domain of f = domain of g i.e B = C.

In our case, the co-domain of g is R and the domain of f is also R  so (fog) (x) is defined.

Composition of function:

(fog) (x) = f[g(x)]

Calculations:

Given: f, g : R → R be two functions defined as f(x) = |x| + x and g(x) = |x| - x ∀ x ∈ R

Given, x

If(x) = |x| + x and g(x) = |x| - x 

Now, (fog) (x) = f[g(x)]

= |g(x)| + g(x)

For x

⇒ (fog) (x) = f[g(x)] = |g(x)| + g(x)

= 0 + 0 = 0

Concept:

• For two functions f(x) and g(x), (f o g)(x) is defined as f[g(x)].

Calculation:

f(x) = sin x and g(x) = x2.

∴ (f o g)(x) = f[g(x)] = sin [g(x)] = sin (x²) = sin x².

Concept:

For any two functions f and g, f o g is defined as f[g(x)].

Calculation:

Given that,

f(x) = 4x + 3   

Using the above concept

⇒ fof(x) = 4(4x + 3) + 3

⇒ fof(x) = 16x + 15 

Again using the same concept

⇒ fofof(x) = 16(4x + 3) + 15

⇒ fofof(x) = 64x + 63

Put x = -1 in the above function

⇒ fofof(-1) = 64(-1) + 63 = -1

∴  f o f o f(-1) equal to -1.

Concept: 

Composition of function: Let f, and g be any functions then, f ∘ g(x) = f[g(x)]

Calculation:

Here, f(x) = 

f{f(x)} = 

= 

= 

= x

Hence, option (3) is correct.

Concept: 

Composition of function: Let f, and g be any functions then, f ∘ g(x) = f[g(x)]

Calculation:  

Given f (x) = 16x⁴ and g (x) = x^1/4 

gof (x) = g[f(x)] =  

⇒ gof (x) =  

⇒ gof (x) = 2x . 

The correct option is 4. 

Concept: 

Composition of function: Let f, and g be any functions then, f ∘ g(x) = f[g(x)]

Calculation:  

Given f (x) = 8x3 and g (x) = x1/3 

gof (x) = g[f(x)] =  

⇒ gof (x) =  

⇒ gof (x) = 2x . 

Now put x = 2

So, gof (2) = 4 . 

The correct option is 2. 

Concept: 

Composition of function: Let f, and g be any functions then, f ∘ g(x) = f[g(x)]

Calculations:

Given, f(x) = e^x and g(x) = loge x

Now, f ∘ g(x) = f[g(x)]

= e^g(x)

= e^loge x

= x                (∵ e^loge x = x)

f ∘ g(x) = x

Put x = 1, we get

fog(1) = 1

Concept:

Composition of function: fog(x) = f(g(x)), it means that the value of x for the function f(x) is g(x).

Example: Suppose f(x ) = x + 2 and g(x) = 2x. Then f(g(x)) = g(x) + 2 = 2x + 2

Calculation:

Given that, f(x) =  and g(x) =

f[g(x)] =

Hence, option (4) is correct.

Mistake Points
For option C, the Numerator should be (4x + x2 + 4) but it is given as (2x + x2 + 4). Therefore, option 4 is correct answer.

Calculation:

Given: f(x) = px + q and g(x) = mx + n

f (g(x)) = g (f(x))

⇒ f (mx + n) = g (px + q)

⇒ p (mx + n) + q = m (px + q) + n

⇒ pmx + pn + q = pmx + mq + n

⇒ pn + q = mq + n

⇒ f (n) = g (q)

∴ Option 3 is correct answer.

Calculation:

Here, f(x) + 2f() =                 ....(1)

Replace x by , we get 

f() + 2f(x) = x                          ....(2)

Adding (1) and (2), we get  

 3f(x) + 3f()) = x +                ....(3)

Now, subtracting (1) from (2), we get

f(x) - f() = x - 

Multiplying by 3,

3f(x) - 3f() = 3x -                   ....(4)

Now, adding (3) and (4) we get 

6f(x) = 4x - 

f(x) = 

Hence, option (4) is correct.