# FIND THE FUNCTIONS F AND G FROM THE COMPOSITION FUNCTION

In (f ∘ g) (x),

(f ∘ g) (x) = f[g(x)]

Here g(x) is the input for the function f(x).

Wherever we see x in the function f(x), we replace it by the function g(x).

Example :

(f ∘ g) (x) = 1/(x - 3)

Solution :

In the given composition function, in the place of x, we see x - 3.

Since x - 3 is the replacement, g(x) = x - 3

Then,

f(x) = 1/x

Finding the composition of function (f ∘ g) (x), the result matches with H(x). So, the required functions are

f(x) = 1/x and g(x) = x - 3

Note :

We can find more than one possible functions for the given composition of function.

Find functions f and g so that f∘g = H

Problem 1 :

H(x) = (2x + 3)4

Solution :

Given that,

(f∘g)(x) = H (x)

f [ g(x) ]

From this, we can understand that where we see x in the function f(x), it is being replaced by the function g(x).

Let g(x) = 2x + 3, then f(x) must be x4

Verifying our assumption :

= f(2x + 3)

= (2x + 3)4

So, the required functions f and g are

f(x) = x4 and g(x) = 2x + 3

Problem 2 :

H(x) = (1+x2)3

Solution :

Given that,

(f∘g)(x) = H (x)

f [ g(x) ]

One possible answer :

If g(x) = 1 + x2, then  f(x) = x3

Other possible answers are also there :

If g(x) = x2, then f(x) = (1 + x)3

Problem 3 :

H(x) = √(x2 + 1)

Solution :

Given that,

(f∘g)(x) = H (x)

f [ g(x) ]

One possible answer :

If g(x) = √(x + 1), then  f(x) = x2

Other possible answer :

If g(x) = √x, then  f(x) = x2 + 1

Problem 4 :

H(x) = |2x + 1|

Solution :

Given that,

(f∘g)(x) = H (x)

f [ g(x) ]

One possible answer :

If g(x) = |x|, then  f(x) = 2x + 1

One possible answer :

If g(x) = |x - 1|, then  f(x) = 2(x + 1)

Problem 5 :

H(x) = |2x2 + 3|

Solution :

Given that,

(f∘g)(x) = H (x)

f [ g(x) ]

One possible answer :

If g(x) = |2x+3|, then  f(x) = x2

Problem 6 :

Most functions are compositions of basic functions. Work backwards to determine the basic functions that created the composition

Solution :

Let H(x) = f(g(x))

a) f(g(x)) = (x + 4)2 + 5

H(x) = (x + 4)2 + 5

If f(x) = x2 + 5 and g(x) = x + 4

b) f(g(x)) = √(x - 4)

H(x) = √(x - 4)

If g(x) = x - 4, then f(x) = √x

c) f(g(x)) = (4x - 1)2

H(x) = (4x - 1)2

If g(x) = 4x - 1, then f(x) = x2

d) f(g(x)) = |x + 2|

H(x) = |x + 2|

If g(x) = (x + 2), then f(x) = |x|

e) f(g(x)) = √x - 4

H(x) = √x - 4

If g(x) = √x, then f(x) = x - 4

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