Domain and range of f(x) = 4/(x + 1) - 2

Step by step

The question

Question: find the domain and range of f(x) = 4/(x + 1), − 2.

The domain is all the x's we can put in.

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

1 Find where the bottom is 0

We can never divide by 0.

So the bottom can never be 0.

The bottom x + 1 is 0 when x = −1: −1 plus 1 is 0.

So the domain is all real numbers except −1: x ≠ −1.

Now the range: which y's can come out?

2 What a fraction can be

A fraction is 0 only when its top is 0.

Here the top is 4.

So 4/(x + 1) is never 0.

It can be any other number.

Careful!

−1 is the x that can't go in.

The range is about the y's that come out.

So 4/(x + 1) ≠ 0.

3 Take away the 2

In f(x), − 2 comes after.

So we take away 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.

4 See it on the graph

Draw the grid.

Here is the graph of f(x) = 4/(x + 1), − 2.

It never touches x = −1: that x can't go in.

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

5 Check

Let's check.

Any other y can come out: x = 0 gives y = 2.

But y = −2 would need 4/(x + 1) = 0, and the top 4 is never 0.

So −2 never comes out.

Let's look back.

First: find where the bottom is 0.

Then: what a fraction can be.

Then: take away the 2.

Then: see it on the graph.

Last: check.