Composite Functions

Question 1

In plainspeak,
and in
one word,
functions are _____.

Answer 1

Maps

They map from
one set of numbers to
another set.

Question 2

Given functions
g(x) and h(n),
how do you describe
g(h(n))?

Answer 2

A
composite function

Question 3

Formally,
what is the
definition of a
function composition?

Answer 3

( M ◦ C )(a) = M (C (a))

”( M ◦ C )(a)” is read as
“M composed with C”

◦
is the
composite function symbol

Question 4

In plainspeak,
what do
composite functions do?

Answer 4

Take the
input of one function
and give the
output of another function

Question 5

If

g(x) = x³
and
h(n) = 4n − √(n),

what is
(h ◦ g)?

Answer 5

4x³ − √(x³)

Plug in h(n) as the input value of g(x).

Work from the
inside out.

• Can simplify, but is not necessary.*
• Could also be written as h(g(x)).*

Question 6

If

g(x) = x³
and
h(n) = 4n − √(n),

what is
(g ◦ h)?

Answer 6

(4n − √(n))³

Plug in h(n) as the input value of g(x).

Work from the
inside out.

• Can simplify, but is not necessary.*
• Could also be written as g(h(n)).*

Question 7

Given:

• For function D
  
  • Input: Speed of the ship, k
  • Output: Duration of the cruise, D(k)
• For function S
  
  • Input: Rate of the river current, x
  • Output: Speed of the ship, S(x)

In words,
what will
D(S(x)) tell you?

Answer 7

The
duration of the cruise
as a function of
the rate of the river current

Question 8

What is
h (f(2))?

Answer 8

6

Question 9

What is
f (h(3))?

Answer 9

−11

Question 10

Given
f(x) in blue
and
g(x) in orange,

what best approximates
(g ◦ f)(0)?

Answer 10

1

Question 11

Given
f(x) in blue
and
g(x) in orange,

what best approximates
(f ◦ g)(2)?

Answer 11

3

Question 12

Formally, if

• *functions f and g** are
• *inverses**, what
• *conditions** must be true?

Answer 12

f (g (x)) = x for all x in the domain of g

g (f (x)) = x for all x in the domain of f

Composing f and g,
in either order,
creates the function that
for every input
returns that input.

Question 13

Given that

• *f(x) and g(x)** are
• *inverse functions**,

f(g(17)) = _____?

Answer 13

17

For inverse functions f(x) and g(x),
f(g(x)) = x = g(f(x))

Question 14

f is a
finite function
that is defined by the table below.

Is f an
invertible function?

How do you know?

Answer 14

Yes.

• *Every member** of the domain
• *corresponds** to a
• *unique member** of the range.

Question 15

h is a
finite function
that is defined by the
mapping diagram below.

Is h an
invertible function?

How do you know?

Answer 15

No.

• *Not every member** of the domain
• *corresponds** to a
• *unique member** of the range.

See diagram below.
h⁻¹ is not a function.

Question 16

g is a
finite function
that is defined by the table below.

Is g an
invertible function?

How do you know?

Answer 16

No.

• *Not every member** of the domain
• *corresponds** to a
• *unique member** of the range.

Question 17

y = x³
is
graphed below.

1. Is it
   a function?
2. If so, is it
   invertible?
3. From visual inspection alone,
   how do you know
   whether it’s invertible?

Answer 17

Yes, it’s a function.

Yes, it’s invertible.

It
passes the horizontal line test.

Question 18

y = x²
is
graphed below.

1. Is it
   a function?
2. If so, is it
   invertible?
3. From visual inspection alone,
   how do you know
   whether it’s invertible?

Answer 18

Yes, it’s a function.

No, it’s not invertible.

It’s not invertible because it
fails the horizontal line test.

Question 19

Given that

f(x) = 2x + 7
and
h(x) = x − 7
2

how do you determine
whether f(x) and h(x) are
**inverse fuctions**?

Answer 19

Write simplified expressions for
f(h(x)) and h(f(x))
in terms of x.

Example:

STEP ONE:

• f(x) = 2x + 7
• f(h(x)) = 2(h(x)) + 7 (DO NOT SKIP THIS STEP)
• = 2 • x − 7 + 7
  2
• = x − 7 ​+ 7
• = x

STEP TWO:

• h(x) = x − 7
  2
• h(f(x)) = f(x) − 7 (DO NOT SKIP THIS STEP)
  2
• = 2x + 7 − 7
  2
• = 2x
  2
• = x

f(h(x)) = x = h(f(x))