Question

What’s is the range of the function

Answer 1

Given:

`$f(x)=(8 x-3)/(8 x+3)`

To find:

The range of the function.

Solution:

The range of a function is the complete set of all possible resulting values of the dependent variable.

The function range is combined domain of the inverse functions.

Let us find the inverse of the function.

`$y=(8 x-3)/(8 x+3)`

Interchange the variables x and y.

`$x=(8 y-3)/(8 y+3)`

Multiply both sides by 8y + 3.

`x(8y+3)=8y-3`

`8xy+3x=8y-3`

Arrange y terms one side to solve to y.

`8 x y-8 y=-3-3 x`

`8 y(x-1)=-3-3 x`

Divide by 8(x - 1) on both sides, we get

`$y=(-3-3 x)/(8(x-1))`

The domain of a function is the set of input or argument values for which the function is real and defined .

At x = 1, the function is undefined.

The function domain is x < 1 or x > 1.

Combining intervals, we get f(x) < 1 or f(x) > 1.

Therefore the range of the function is
`(-\infty, 1) \cup(1, \infty)`.