Domain and range of f(x) = -1/x + 3

Step by step

The question

Question: find the domain and range of f(x) = −1/x, + 3.

The domain is all the x's we can put in.

The range is all the y's that can come out.

1 Find where the bottom is 0

We can never divide by 0.

So the bottom can never be 0.

The bottom is just x.

So x = 0 makes it 0.

So the domain is all real numbers except 0: x ≠ 0.

Now the range: which y's can come out?

2 What a fraction can be

A fraction is 0 only when its top is 0.

Here the top is −1.

So −1/x is never 0.

It can be any other number.

Careful!

0 is the x that can't go in.

The range is about the y's that come out.

So −1/x ≠ 0.

3 Add the 3

In f(x), + 3 comes after.

So we add 3.

We do it on both sides, so the sign stays true.

The left side is f(x), the y.

So y ≠ 3.

Every y except 3 comes out.

4 See it on the graph

Draw the grid.

Here is the graph of f(x) = −1/x, + 3.

It never touches x = 0: that x can't go in.

And it never touches y = 3: that y never comes out.

5 Check

Let's check.

Any other y can come out: x = 1 gives y = 2.

But y = 3 would need −1/x = 0, and the top −1 is never 0.

So 3 never comes out.

Let's look back.

First: find where the bottom is 0.

Then: what a fraction can be.

Then: add the 3.

Then: see it on the graph.

Last: check.