Solve f(x) = x^2 + 1, g(x) = 2x - 1, (f∘g)(x)

Step by step

The question

Question: find f(g(x)) for f(x) = x² + 1 and g(x) = 2x − 1.

1 Work out the inside

f(g(x)) is a function inside a function.

Like with brackets, we start inside.

First g(x).

g(x) is 2x − 1.

So f(g(x)) is f(2x − 1).

2 Put it in

Put (2x − 1) in place of every x in f.

We keep it in brackets.

3 Write the square out

A square means the bracket times itself.

(2x − 1)² is 2x − 1 times 2x − 1.

Careful!

We can't just square each part.

4x² + 1 is wrong.

The middle part is missing.

So we write the bracket two times.

4 Split the first bracket

Each part of the first bracket multiplies the whole second bracket.

The parts of the first bracket are 2x and −1.

Each one takes the whole second bracket.

5 Open the brackets

What is in front of a bracket multiplies every part inside.

So we write each part times what is in front.

Watch the signs.

2x times 2x is 4x².

2x times (−1) is −2x.

(−1) times 2x is −2x.

(−1) times (−1) is 1.

6 Group like terms

Terms are the parts between the + and − signs.

Like terms are the same kind: same letters, same powers.

All the x terms go together.

Plain numbers go together.

Each colour is one kind.

−2x − 2x is −4x.

1 + 1 is 2.

So the expression is 4x² − 4x + 2.

7 Check

Let's check.

We pick a number for each letter: x = 2.

We put it into the start.

Now into our answer.

Both give 10.

Same value, so 4x² − 4x + 2 is right!

Let's look back.

First: work out the inside.

Then: put it in.

Then: write the square out.

Then: split the first bracket.

Then: open the brackets.

Then: group like terms.

Last: check.