Range of f(x) = 1/(x + 2) - 1 Step by Step

Step by step

The question

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

f(x) = 1x + 2 − 1

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

range = 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 + 2) is never 0.

It can be any other number.

a fraction is 0 only when its top is 0

the top is 1: never 0

Careful!

−2 is the x that can't go in.

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

Careful: not range: y ≠ −2 ✗

So 1/(x + 2) ≠ 0.

1x + 2 ≠ 0

2 Take away the 1

In f(x), − 1 comes after.

So we take away 1.

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

1x + 2 − 1 ≠ 0 − 1

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

So y ≠ −1.

Every y except −1 comes out.

y ≠ −1

3 See it on the graph

Draw the grid.

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

And it never touches y = −1: that y never comes out.

4 Check

Let's check.

Any other y can come out: x = −1 gives y = 0.

x = −1: 1−1 + 2 − 1 = 1 − 1 = 0

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

So −1 never comes out.

y = −1: 1x + 2 = 0 never true

Let's look back.

First: what a fraction can be.

Then: take away the 1.

Then: see it on the graph.

Last: check.

Answer