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

Step by step

The question

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

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 x − 3 is 0 when x = 3: 3 minus 3 is 0.

So the domain is all real numbers except 3: x ≠ 3.

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 − 3) is never 0.

It can be any other number.

Careful!

3 is the x that can't go in.

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

So 1/(x − 3) ≠ 0.

3 Add the 1

In f(x), + 1 comes after.

So we add 1.

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

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

So y ≠ 1.

Every y except 1 comes out.

4 See it on the graph

Draw the grid.

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

It never touches x = 3: that x can't go in.

And it never touches y = 1: that y never comes out.

5 Check

Let's check.

Any other y can come out: x = 4 gives y = 2.

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

So 1 never comes out.

Let's look back.

First: find where the bottom is 0.

Then: what a fraction can be.

Then: add the 1.

Then: see it on the graph.

Last: check.