f(x) = x^2 + 1, g(x) = x^2 - 4x, find (f + g)(x) Step by Step

Step by step

The question

Question: find (f + g)(x) for f(x) = x² + 1 and g(x) = x² − 4x.

f(x) = x² + 1

g(x) = x² − 4x

(f + g)(x) means f of x plus g of x.

We add the two answers.

(f + g)(x) = f(x) + g(x)

1 Put the rules in

Each rule goes in brackets, as one piece: (x² + 1) + (x² − 4x).

(f + g)(x) = (x² + 1) + (x² − 4x)

2 Open the brackets

A bracket with a plus or nothing in front can just be removed.

( ) away

Now we have x² + 1 + x² − 4x.

= x² + 1 + x² − 4x

3 Group like terms

Terms are the parts between the + and − signs.

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

All the x² terms go together.

Each colour is one kind.

like terms = same letters,

same powers

We move like terms next to each other.

Each term takes its sign with it.

= x² + x² − 4x + 1

x² + x² is 2x².

It is like 1 apple plus 1 apple: that is 2 apples.

x² + x² = 2x²

So the expression is 2x² − 4x + 1.

(f + g)(x) = 2x² − 4x + 1

4 Check

Let's check with a number: x = 2.

f of 2 is 5, g of 2 is −4.

Plus: 1.

f(2) + g(2) = 5 + (−4) = 1

Now our answer with x = 2.

We get 1 too.

It works!

Answer: 2(2)² − 4(2) + 1 = 8 − 8 + 1 = 1

Let's look back.

First: put the rules in.

Then: open the brackets.

Then: group like terms.

Last: check.

Answer