Range of f(x) = 1/x

Step by step

The question

Question: find the range of f(x) = 1 over x.

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!

The domain is the x's.

The range is the y's.

We look at what comes out.

So 1/x ≠ 0.

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

So y ≠ 0.

Every y except 0 comes out.

2 See it on the graph

Draw the grid.

Here is the graph of f(x) = 1 over x.

And it never touches y = 0: that y never comes out.

3 Check

Let's check.

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

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

So 0 never comes out.

Let's look back.

First: what a fraction can be.

Then: see it on the graph.

Last: check.