f(x) = x + 6, g(x) = -2x, 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) = x + 6 and g(x) = −2x.

f(x) = x + 6

g(x) = −2x

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 −2x.

So f(g(x)) is f(−2x).

f(g(x)) = f(−2x)

Put (−2x) in place of every x in f.

We keep it in brackets.

= (−2x) + 6

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

( ) away

Now we have −2x + 6.

f(g(x)) = −2x + 6

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 x + 6.

So g(f(x)) is g(x + 6).

g(f(x)) = g(x + 6)

Put (x + 6) in place of every x in g.

We keep it in brackets.

= −2(x + 6)

The (−2) in front of the bracket multiplies every part inside.

a(b + c) = ab + ac

So we write (−2) times each part.

= (−2)(x) + (−2)(6)

− × + = −

Watch the signs.

(−2) times x is −2x.

(−2) times 6 is −12.

g(f(x)) = −2x − 12

3 Compare

−2x + 6 and −2x − 12 are not the same.

So the order matters!

−2x + 6 ≠ −2x − 12

Careful!

Inside first, then outside.

Swap them, and you usually get a different rule.

Careful: f(g(x)) is usually not g(f(x))

4 Check

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

g of 1 is −2, then f of −2 is 4.

f(g(1)) = f(−2) = 4

f of 1 is 7, then g of 7 is −14.

Different, as we found.

g(f(1)) = g(7) = −14

And our rules give the same numbers: −2x + 6 gives 4, −2x − 12 gives −14.

They are right!

−2x + 6 → 4, −2x − 12 → −14

Let's look back.

First: find f(g(x)).

Then: find g(f(x)).

Then: compare.

Last: check.

Answer