Find the domain and the range of each of the following real function:

1.	Find the domain and the range of each of the following real function: f(x) = \(\frac{3x-2}{x+2}\)
Answer» Given: f(x) = \(\frac{3x-2}{x+2}\) Need to find: Where the functions are defined. Let, f(x) = \(\frac{3x-2}{x+2}\) = y....(1) To find the domain of the function f(x) we need to equate the denominator of the function to 0. Therefore, x + 2 = 0 ⇒ x = -2 It means that the denominator is zero when x = -2 So, the domain of the function is the set of all the real numbers except -2. The domain of the function, D_f(x) = (- ∞, -2) ∪ (-2, ∞). Now, to find the range of the function we need to interchange x and y in the equation no. (1) So the equation becomes, \(\frac{3y-2}{2+y}\) = x ⇒ 3y - 2 = 2x + xy ⇒ 3y - xy = 2x + 2 ⇒ y = \(\frac{2x+2}{3-x}\) = f(x₁) To find the range of the function f(x₁) we need to equate the denominator of the function to 0. Therefore, 3 – x = 0 ⇒ x = 3 It means that the denominator is zero when x = 3 So, the range of the function is the set of all the real numbers except 3. The range of the function, R_f(x) = (- ∞, 3) ∪ (3, ∞).

Find the domain and the range of each of the following real function: f(x) = \(\frac{3x-2}{x+2}\)

Answer»

Given: f(x) = \(\frac{3x-2}{x+2}\)

Need to find: Where the functions are defined.

Let, f(x) = \(\frac{3x-2}{x+2}\) = y....(1)

To find the domain of the function f(x) we need to equate the denominator of the function to 0.

Therefore,

x + 2 = 0

⇒ x = -2

It means that the denominator is zero when x = -2

So, the domain of the function is the set of all the real numbers except -2.

The domain of the function, D_f(x) = (- ∞, -2) ∪ (-2, ∞).

Now, to find the range of the function we need to interchange x and y in the equation no. (1)

So the equation becomes,

\(\frac{3y-2}{2+y}\) = x

⇒ 3y - 2 = 2x + xy

⇒ 3y - xy = 2x + 2

⇒ y = \(\frac{2x+2}{3-x}\) = f(x₁)

To find the range of the function f(x₁) we need to equate the denominator of the function to 0.

Therefore,

3 – x = 0

⇒ x = 3

It means that the denominator is zero when x = 3

So, the range of the function is the set of all the real numbers except 3.

The range of the function, R_f(x) = (- ∞, 3) ∪ (3, ∞).