# How do you find the domain and range of #f(x)=2/(3/(x-6))-7#?

##### 1 Answer

Starting with a domain and range of the set of all real numbers, exclude domain values for which division by zero may occur, and exclude range values that would result from the forbidden domains, if no other domain maps to said range values. This yields:

Domain:

Range:

#### Explanation:

Our function is

Since there are no radicals (where the imaginary number may reside), we could safely assume that the domain and range is the set of all real numbers...

... But we need to **remember the wrath of division by zero**.

To prevent any divisions by zero, we need to make sure that the denominator cannot equal zero. In this function, there is a fraction

where the denominator

cannot equal zero:

However, inside this denominator, there is also a denominator

which also cannot equal zero:

To figure out what the forbidden numbers are, we could solve for

But how about that bigger denominator?

Well, we could first try to multiply by the smaller denominator...?

But wait! We need to make sure

Well, of course

Domain:

And the range? Is it all real numbers too? Take a moment to think about it. If

Hmm... we could start by rewriting the fraction as plain division:

Division should "flip" the numerator and denominator (due to being the multiplicative inverse):

We could multiply the

And "split" the

Leaving us to simplify:

Nice, this is a linear function! Since its graph is a line, each range should also be associated to one and only one domain. With no divisions by

And solve:

Ah! So that value,

Domain:

Range: