Range of f(x) = 1/x + 2

Step by step

The question

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

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) = 1/x, + 2.

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

4 Check

Let's check.

Any other y can come out: x = 1 gives y = 3.

But y = 2 would need 1/x = 0, and the top 1 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.