**How to find range of a function without graphing ?**

Here we are going to see how to find range of a function without graphing.

**What is range ?**

The range of real function of a real variable is the step of all real values taken by f(x) at points in its domain.

To find the range of the real function, we need to follow the steps given below.

**Step 1 :**

Put y = f(x)

**Step 2 :**

Solve the equation y = f(x) for x in terms of y. Let x = g(y)

**Step 3 :**

Find the values of y for which the values of x, obtained from x = g(y) are real and its domain of f.

**Step 4 :**

The set of values of y obtained in step 3 is the range of the given function.

Let us look into some example problems to understand the above concept.

**Example 1 :**

Find the range of the following function

f(x) = (4 - x) / (x - 4)

**Solution :**

y = (4 - x) / (x - 4)

y = -(x - 4) / (x - 4)

y = -1

Hence the range of f(x) is -1

**Example 2 :**

Find the range of the following function

f(x) = (x^{2} - 9) / (x - 3)

**Solution :**

y = (x^{2} - 9) / (x - 3)

Domain of f(x) is R - {3}

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

y = x + 3

x = y - 3

Hence the range of f(x) is R - {6}

**Example 3 :**

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}

**Example 4 :**

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, ∞)

**Example 5 :**

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 "How to find range of a function without graphing".

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