**Find composition of two functions :**

Here we are going to see how to find composition of two functions.

We follow the steps given below to find the composition of two functions f with g

**Step 1 :**

If f(x) = x and g(x) = 3x^{2}

fₒ g (x) = f [ g(x) ]

**Step 2 :**

In this step we have to apply the value 3x^{2 }instead of x in the function f (x).

f(3x^{2}) = 3x^{2}

**Step 3 :**

The answer that we got in step 2 represents the composition of two functions. If it is possible we have to simplify the answer in step 2.

That is,

fₒ g (x) = 3x^{2}

Let us look into some examples to understand the above concept.

**Example 1 :**

If f(x) = -4x + 2 and g(x) = √(x- 8) find fₒ g (x)

**Solution :**

Given : f(x) = -4x + 2 and g(x) = √(x- 8)

**Step 1 :**

fₒ g (x) = f [g(x)]

By applying the value of g(x) in the above step, we get

= f [ √(x- 8)]

**Step 2 :**

Now we have to apply x = √(x- 8) in the function of f (x).

= f(√(x- 8)) = -4√(x- 8) + 2

**Step 3 :**

We cannot simplify -4√(x- 8) + 2 here after.

Hence the value of fₒ g (x) is -4√(x- 8) + 2

**Example 2 :**

If f(x) = -3x + 4 and g(x) = x^{2} find g ₒ f (x)

**Solution :**

Given : f(x) = -3x + 4 and g(x) = x^{2}

g ₒ f (x) = g [f(x)]

= g (-3x + 4)

Here x = -3x + 4. Now we are going to apply this value in the function g(x).

g(-3x + 4) = (-3x + 4)^{2}

We may expand this using algebraic identity

= (-3x)^{2} + 2 (-3x) (4) + 4^{2}

= 9x^{2} - 24x + 16

Hence the value of g ₒ f (x) is 9x^{2} - 24x + 16

**Example 3 :**

If f(x) = 2x - 5 and g(x) = x + 2 find f ₒ g (x)

**Solution :**

Given : f(x) = 2x - 5 and g(x) = x + 2

f ₒ g (x) = f [g(x)]

= f [x + 2]

In the function f(x), we have to apply x + 2 instead of x.

f(x + 2) = 2(x + 2) - 5

= 2x + 4 - 5

= 2x - 1

Hence the value of f ₒ g (x) is 2x - 1.

**Example 4 :**

If f(x) = -9x + 3 and g(x) = x^{4} find g ₒ f (x)

**Solution :**

Given : f(x) = -9x + 3 and g(x) = x^{4}

g ₒ f (x) = g [f(x)]

= g [-9x + 3]

In the function g(x), we have to apply -9x + 3 instead of x.

g(-9x + 3) = (-9x + 3)^{4}

Hence the value of g ₒ f (x) is (-9x + 3)^{4}

After having gone through the stuff given above, we hope that the students would have understood "Find composition of two functions".

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