Question

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Answer 1

Domain and Range of Functions

Given a function y = f(x), the domain of f is

The range of f is the

We are given the function:

`f(x)=(1)/(2x)-3`

It's required to find the domain and range of the inverse function.

We must recall that, if the inverse function exists, its domain is the range of f, and its range is the domain of f.

Since f(x) is a rational function, it may not exist for the values of x where the denominator is 0.

Concretely, for x=0, the denominator is 0 and the function does not exist, thus the domain of f consists of every real number except x=0.

This can be written as (-∞,0) U (0,∞). This is also the range of the inverse function.

Now let's find the inverse function. We must solve for x the equation:

`y=(1)/(2x)-3`

Adding 3:

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

Multiplying by x and dividing by y+3:

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

Swapping letters:

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

This is the inverse function of f.

Thus the domain of the inverse is:

-3-3

And the range:

Answer: Choice d