HOW TO FIND RANGE OF A FUNCTION

Find the range of the following functions :

Example 1 :

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

Solution :

Let y = (x - 2)/(3 - x).

Solve for x.

y = (x - 2)/(3 - x)

Multiply both sides by (3 - x).

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

3y - xy = x - 2

Add xy and 2 to both sides.

3y + 2 = x + xy

3y + 2 = x(1 + y)

Divide both sides by (1 + y).

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

In the above equation, the denominator y + 1 will become zero, if y = -1.

So, x is undefined when y = -1.

Hence the range is R - {-1}.

Example 2 :

f(x) = 1/√(x - 5)

Solution :

Let y = 1/√(x - 5).

y = 1/√(x - 5)

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

√(x - 5) > 0  ==> 1/√(x - 5) > 0

Thus y takes all real values greater than zero.

Hence range of f(x) is (0, +∞).

Example 3 :

f(x) = √(16 - x²)

Solution :

y = √(16 - x²)

Squaring both sides,

y² = 16 - x²

x² = 16 - y²

x = √(16 - y²)

Clearly x will take all real values, if

(16 - y²) ≥ 0  ==> y² - 16 ≤ 0  ==> (y + 4)(y - 4) ≤ 0

-4 ≤ y ≤ 4  ==>  y ∈ [-4, 4]

Also, y = √(16 - x²) ≥ 0 for all x ∈ [-4, 4].

Thus y ∈ [0, 4] for all x ∈ [-4, 4].

Hence the range is [0, 4].

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