Range of f(x) = 2/x - 3

Step by step

The question

Question: find the range of f(x) = 2/x, − 3.

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 2.

So 2/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 2/x ≠ 0.

2 Take away the 3

In f(x), − 3 comes after.

So we take away 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.

3 See it on the graph

Draw the grid.

Here is the graph of f(x) = 2/x, − 3.

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

4 Check

Let's check.

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

But y = −3 would need 2/x = 0, and the top 2 is never 0.

So −3 never comes out.

Let's look back.

First: what a fraction can be.

Then: take away the 3.

Then: see it on the graph.

Last: check.