Range of f(x) = -2/(x - 3) + 4

Step by step

The question

Question: find the range of f(x) = −2/(x − 3), + 4.

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 − 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 −2/(x − 3) ≠ 0.

2 Add the 4

In f(x), + 4 comes after.

So we add 4.

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

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

So y ≠ 4.

Every y except 4 comes out.

3 See it on the graph

Draw the grid.

Here is the graph of f(x) = −2/(x − 3), + 4.

And it never touches y = 4: that y never comes out.

4 Check

Let's check.

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

But y = 4 would need −2/(x − 3) = 0, and the top −2 is never 0.

So 4 never comes out.

Let's look back.

First: what a fraction can be.

Then: add the 4.

Then: see it on the graph.

Last: check.