Range of f(x) = 5/(x - 2) + 2

Step by step

The question

Question: find the range of f(x) = 5/(x − 2), + 2.

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

1 What a fraction can be

A fraction is 0 only when its top is 0.

Here the top is 5.

So 5/(x − 2) is never 0.

It can be any other number.

Careful!

The domain is the x's.

The range is the y's.

We look at what comes out.

So 5/(x − 2) ≠ 0.

2 Add the 2

In f(x), + 2 comes after.

So we add 2.

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

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

So y ≠ 2.

Every y except 2 comes out.

3 See it on the graph

Draw the grid.

Here is the graph of f(x) = 5/(x − 2), + 2.

And it never touches y = 2: that y never comes out.

4 Check

Let's check.

Any other y can come out: x = 3 gives y = 7.

But y = 2 would need 5/(x − 2) = 0, and the top 5 is never 0.

So 2 never comes out.

Let's look back.

First: what a fraction can be.

Then: add the 2.

Then: see it on the graph.

Last: check.