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

Step by step

The question

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

f(x) = x² + 2

g(x) = x + 1

1 Find f(g(x))

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

Like with brackets, we start inside.

First g(x).

inside first: g(x)

g(x) is x + 1.

So f(g(x)) is f(x + 1).

f(g(x)) = f(x + 1)

Put (x + 1) in place of every x in f.

We keep it in brackets.

= (x + 1)² + 2

A square means the bracket times itself.

(x + 1)² is x + 1 times x + 1.

( )² = ( )( )

Careful!

We can't just square each part.

x² + 1 is wrong.

The middle part is missing.

Careful: not (x + 1)² = x² + 1 ✗

So we write the bracket two times.

= (x + 1)(x + 1) + 2

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

(a + b)(c + d) = a(c + d) + b(c + d)

The parts of the first bracket are x and 1.

Each one takes the whole second bracket.

= x(x + 1) + 1(x + 1) + 2

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

a(b + c) = ab + ac

So we write each part times what is in front.

= x(x) + x(1) + 1(x) + 1(1) + 2

x times x is x².

x times 1 is x.

1 times x is x.

1 times 1 is 1.

= x² + x + x + 1 + 2

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.

like terms = same letters,

same powers

x + x is 2x.

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

1 + 2 is 3.

x + x = 2x

1 + 2 = 3

So the expression is x² + 2x + 3.

f(g(x)) = x² + 2x + 3

2 Find g(f(x))

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

Like with brackets, we start inside.

First f(x).

f(x) is x² + 2.

So g(f(x)) is g(x² + 2).

g(f(x)) = g(x² + 2)

Put (x² + 2) in place of every x in g.

We keep it in brackets.

= (x² + 2) + 1

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

( ) away

Now we have x² + 2 + 1.

= x² + 2 + 1

Terms are the parts between the + and − signs.

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

Plain numbers go together.

Each colour is one kind.

like terms = same letters,

same powers

2 + 1 is 3.

2 + 1 = 3

So the expression is x² + 3.

g(f(x)) = x² + 3

3 Compare

x² + 2x + 3 and x² + 3 are not the same.

So the order matters!

x² + 2x + 3 ≠ x² + 3

4 Check

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

g of 1 is 2, then f of 2 is 6.

f(g(1)) = f(2) = 6

f of 1 is 3, then g of 3 is 4.

Different, as we found.

g(f(1)) = g(3) = 4

And our rules give the same numbers: x² + 2x + 3 gives 6, x² + 3 gives 4.

They are right!

x² + 2x + 3 → 6, x² + 3 → 4

Let's look back.

First: find f(g(x)).

Then: find g(f(x)).

Then: compare.

Last: check.

Answer