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

Find the domain and the range of each of the following real function: f(x) = \(\frac{1}x\)

Answer»

Given: f(x) = \(\frac{1}x\)

Need to find: Where the functions are defined.

Let, f(x) = \(\frac{1}x\) = y.......(1)

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

Therefore,

x = 0

It means that the denominator is zero when x = 0

So, the domain of the function is the set of all the real numbers except 0.

The domain of the function, D_f(x) = (- ∞, 0) ∪ (0, ∞).

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

So the equation becomes,

\(\frac{1}y\) = x

⇒ y = \(\frac{1}x\) = f(x₁)

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

Therefore,

x = 0

It means that the denominator is zero when x = 0

So, the range of the function is the set of all the real numbers except 0.

The range of the function, R_f(x) = (- ∞, 0) ∪ (0, ∞).