**Range of function worksheet :**

Here we are going to see some practice questions on range of a function.

(1) Find the range of the following function

f(x) = (x - 2) / (3 - x)

(2) Find the range of the following function

f(x) = 1 / √(x - 5)

(3) Find the range of the following function

f(x) = √(16 - x^{2})

**Question 1 :**

Find the range of the following function

f(x) = (x - 2) / (3 - x)

**Solution :**

y = (x - 2) / (3 - x)

Multiply both sides by (3 - x)

y (3 - x) = (x - 2)

3y - xy = x - 2

In order to solve for x, we need to group x terms

Add 2 on both sides

3y + 2 - xy = x

Add xy on both sides

3y + 2 = x + xy

3y + 2 = x(1 + y)

x = (3y + 2)/(1 + y)

In the denominator we have 1 + y, if we give -1 for y the function will become undefined.

Hence the range is R - { -1}

**Question 2 :**

Find the range of the following function

f(x) = 1 / √(x - 5)

**Solution :**

y = 1 / √(x - 5)

For any x > 5, we have x - 5 > 0

√(x - 5) > 0 ==> 1/√(x - 5) > 0

Thus f(x) takes all real values greater than zero.

Hence range of f(x) is (0, ∞)

**Question 3 :**

Find the range of the following function

f(x) = √(16 - x^{2})

**Solution :**

y = √(16 - x^{2})

Taking squares on both sides, we get

y^{2} = 16 - x^{2}

Add x^{2 }on both sides

x^{2 }= 16 - y^{2}

x = √(16 - y^{2})

Clearly x will take all real values, if

(16 - y^{2}) ≥ 0 ==> y^{2} - 16 ≤ 0 ==> (y + 4) (y - 4) ≤ 0

-4 ≤ y ≤ 4 ==> y ∈ [-4, 4]

Also, y = √(16 - x^{2}) ≥ 0 for all x ∈ [-4, 4].

Thus y ∈ [0, 4] for all x ∈ [-4, 4]

Hence the range is [0, 4]

After having gone through the stuff given above, we hope that the students would have understood "Range of the function worksheet".

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