f(x) = x + 3, g(x) = x - 2, find (f/g)(x) Step by Step

Step by step

The question

Question: find (f/g)(x) for f(x) = x + 3 and g(x) = x − 2.

f(x) = x + 3

g(x) = x − 2

(f/g)(x) means f of x divided by g of x.

We write it as a fraction: f on top, g at the bottom.

(f/g)(x) = f(x)g(x)

1 Put the rules in

Each rule goes in as one piece: (x + 3)/(x − 2).

(f/g)(x) = x + 3x − 2

2 Find what x can't be

We can never divide by 0.

So the bottom, g of x, can never be 0.

÷ 0: not allowed → g(x) ≠ 0

Careful!

The answer is not only the fraction.

x = 2 makes the bottom 0, so we must say it is not allowed.

Careful: write the x that g can't be

Which x makes x − 2 zero?

We solve x − 2 = 0.

x − 2 = 0

3 Remove the − 2

We want x by itself.

The − 2 is in the way.

To remove − 2, we add 2.

We do it on both sides, so they stay equal.

x ~~− 2 + 2~~ = 0 + 2

− 2 and + 2 make zero.

Cross them out.

0 + 2 = 2.

0 + 2 = 2

So x = 2!

x = 2

So (f/g)(x) = (x + 3)/(x − 2), with x ≠ 2.

(f/g)(x) = x + 3x − 2, x ≠ 2

4 Check

Let's check with a number: x = 1.

f of 1 is 4, g of 1 is −1.

Divided: −4.

f(1)g(1) = 4−1 = −4

That is our fraction at x = 1.

It works!

Let's look back.

First: put the rules in.

Then: find what x can't be.

Then: remove the − 2.

Last: check.

Answer