FINDING THE COMPONENTS OF A COMPOSITE FUNCTION

Example 1 :

Find functions f and g such that f o g = H, where

H(x) = (3x - 1)¹²

Solution :

Inside part of the function H(x) is 3x - 1.

We can assume g(x) = 3x - 1 or g(x) = 3x.

Case (i) :

Let g(x) = 3x - 1.

Then, f(x) = x¹².

Check :

f o g = f[g(x)]

= f(3x - 1)

= (3x - 1)¹² ✔

Case (ii) :

Let g(x) = 3x.

Then, f(x) = (x - 1)¹².

Check :

f o g = f[g(x)]

= f(3x)

= (3x - 1)¹² ✔

Example 2 :

Find functions f and g such that f o g = H, where

H(x) = (x³ + 1)⁵⁰

Solution :

Inside part of the function H(x) is x³ + 1.

We can assume g(x) = x³ + 1 or g(x) = x³.

Case (i) :

Let g(x) = x³ + 1.

Then, f(x) = x⁵⁰.

Check :

f o g = f[g(x)]

= f(x³ + 1)

= (x³ + 1)⁵⁰ ✔

Case (ii) :

Let g(x) = x³.

Then, f(x) = (x + 1)⁵⁰.

Check :

f o g = f[g(x)]

= f(x³)

= (x³ + 1)⁵⁰ ✔

Example 3 :

Find functions f and g such that f o g = H, where

H(x) = 1/(5x² + 1)

Solution :

Inside part of the function H(x) is 5x² + 1.

We can assume g(x) = 5x² + 1 or g(x) = 5x².

Case (i) :

Let g(x) = 5x² + 1.

Then, f(x) = 1/x.

Check :

f o g = f[g(x)]

= f(5x² + 1)

= 1/(5x² + 1) ✔

Case (ii) :

Let g(x) = 5x².

Then, f(x) = 1/(x + 1).

Check :

f o g = f[g(x)]

= f(5x²)

= 1/(5x² + 1) ✔

Example 4 :

Find functions f and g such that f o g = H, where

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

Solution :

Inside part of the function H(x) is 2x + 3.

We can assume g(x) = 2x + 3 or g(x) = 2x.

Case (i) :

Let g(x) = 2x + 3.

Then, f(x) = sin(x).

Check :

f o g = f[g(x)]

= f(2x + 3)

= sin(2x + 3) ✔

Case (ii) :

Let g(x) = 2x.

Then, f(x) = sin(x + 3).

Check :

f o g = f[g(x)]

= f(2x)

= sin(2x + 3) ✔

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