**Function composition :**

It is an operation being used to combine the given two functions.

Let f(x) and g(x) be the two functions.

The formula for function composition is given below in different forms.

**f∘g = f[g(x)]**

**f∘g(x) = f[g(x)]**

**f∘g(x²) = f[g(x²)]**

**Problem 1 :**

If f(x) = 2x² + 3 and g(x) = x + 2, find f∘g.

**Solution :**

f∘g = f[g(x)]

f∘g = f[x + 2]

f∘g = 2(x + 2)² + 3

f∘g = 2(x² + 2.x.2 + 2²) + 3

f∘g = 2(x² + 4x + 4) + 3

f∘g = 2x² + 8x + 8 + 3

**f∘g = 2x² + 8x + 11 **

**Problem 2 :**

If f(x) = 5x and g(x) = x+2, find f∘g (x²).

**Solution :**

f∘g (x²) = f[g(x²)]

f∘g (x²) = f[x² + 2]

f∘g(x²) = 5(x² + 2)

**f∘g(x²) = 5x² + 10**

**Problem 3 :**

If f(x) = 5x + 3 and g(x) = 7x - 2, find f∘g(3).

**Solution :**

f∘g(3) = f[g(3)]

f∘g(3) = f[7(3) + 2]

f∘g(3) = f(23)

f∘g(3) = 5(23) + 3

f∘g(3) = 115 + 3

**f∘g(3) = 118**

**Problem 4 :**

If f(x) = x -5 and g(x) = 2x + 3, verify f∘g = g∘f

**Solution :**

f∘g = f[g(x)]

f∘g = f[2x + 3]

f∘g = (2x + 3) - 5

f∘g = 2x + 3 - 5

**f∘g = 2x - 2 ------->(1)**

g∘f = g[f(x)]

g∘f = g[x - 5]

g∘f = 2(x-5) + 3

g∘f = 2x - 10 + 3

**g∘f = 2x - 7 ------->(2)**

**From (1) and (2), it is clear that f∘g ≠ g∘f**

**Problem 5 :**

Let f(x) = x + k and g(x) = 7x. If f∘g(2) = 7, find the value of "k".

**Solution :**

f∘g(2) = 7

f[g(2)] = 7

f[7(2)] = 7

f(14) = 7

14 + k = 7

**k = -7 **

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