Pre-Calculus
Notes Unit 3: Composition of Functions & Inverses
Day 1: Composition of Functions (Algebraically)
Besides using basic operations, there is another way to combine two functions and it is
known as ________________________________.
Basically what happens in composition is that, given two functions, one function
becomes the domain of the other.
Given _______ and _______, “if f(x) is composed with g(x), we have _________________, which
means ________________________________________________________________________________
Examples:
Given the following functions find the requested compositions:
𝑓(𝑥) = 𝑥 2
𝑔(𝑥) = 𝑥 + 1
ℎ(𝑥) = −4𝑥 − 2
𝑠(𝑥) = −2𝑥 2 + 3𝑥 − 1
1) (f ◦ g)(x)
2) (h ◦ g)(x)
3) (s ◦ g)(2)
4) h(f(x))
5) g(f(-1))
6) g(h(0))
Going Backwards:
For what values of "x" is f(g(x)) = 10 if f(x)=2x and g(x)=x+3?
For what values of "x" is f(g(x)) = 12 if f(x)=3x and g(x)=x+2?
Composition of Functions-Graphically
Now, we will use what we know about composition of functions and apply it
_________________________.
Examples:
Given the graph at right, f(x), find the following:
1) f(-2) + f(3) = ___________
2) f(0) – f(4) = __________
3) f(2)f(-2) = _____________
4)
5) f(f(2)) = ______________
𝑓(3)
𝑓(−2)
= ____________
Given two functions f(x) and g(x), use their graphs to answer the following questions:
1) f(0) =
2) f(-3) =
3) f(-1) =
4) g(3) =
5) g(-4) =
6) g(1) =
7) f(g(0)) =
8) g(f(-2)) =
9) f(g(4)) =
10) g(f(2))=
1) f(-2) =
2) f(0) =
3) g(1) =
4) g(-4) =
5) f(g(-2)) =
6) g(f(-4)) =
7) Is it possible for f(g(x)) = -5? Explain.
8) Is it possible for g(f(x)) = 2? Explain.
Day 2: Inverses-Graphing & Algebraically
What is an inverse?
As an example: If I had an ordered pair (-1, 2), its inverse would be (2, -1).
Graphically,
Examples:
Given the following graphs, graph the inverse for each one using a different color (just
different from the original-you don’t need 6 different colors).
Hint: List some easily identifiable ordered pairs, find their inverses and graph the
resulting ordered pairs.
Recall the definitions for relations and functions
Relation:
Example: {(3, 2), (4, 0), (−2, 8), (4, 9)}
Function :
Remember: range values (y) CAN repeat.
Example: {(3, 2), (4, 0), (−2, 8), (−4, 9)}
**When a relation and its inverse are both functions, they are called ___________________.


So, let’s find the inverses for some relations and determine if their inverses are functions.
Examples:
1) {(1, −7), (5, 2), (6, 3), (−2, 0)}
Inverse:
Is the inverse a function?
2) {(6, −7), (8, −2), (9, 0), (−6, 0)} Inverse:
Is the inverse a function?
**So, given an equation, how do you determine its inverse? You switch the x and y, then
solve for y. If the result has a graph that will pass the vertical line test, then the result is
a function.
Examples:
1) 𝑦 =
5−3𝑥
2
Inverse: (pay attention to the notation)
Is the inverse a function?
2) 𝑦 = √𝑥 − 2
Inverse: (pay attention to the notation)
Is the inverse a function?
3) 𝑦 = 𝑥 2 + 4
Inverse: (pay attention to the notation)
Is the inverse a function?
3
4) 𝑦 = √𝑥 − 7
Inverse: (pay attention to the notation)
Is the inverse a function?
Day 3: Finding Inverse Functions
In today’s notes we will learn to find the inverse function of any relation. Since we are
creating a function from the inverse we must now consider restrictions because not all
inverses are automatically functions. We will make the inverses functions by setting
boundaries on the equations.
Horizontal Line Test (HLT):
This test is similar to the Vertical Line Test that enables us to determine if a relation’s
graph is a function, but the HLT is used with inverses. It states that the inverse of a
relation is a function if and only if each horizontal line intersects the graph of the original
relation in at most one point.
This is a good way to check ourselves!
Example:
Examples:
1) 𝑓(𝑥) = 𝑥 2 + 4 𝑓𝑜𝑟 𝑥 ≥ 0
2) 𝑓(𝑥) = √2𝑥 + 5
Proving Inverses by Composition
**
Recall that we can compose functions using:
____________________ remember these can also be written as ______________________
So, how does this work to prove that two functions are inverses of each other?
Given two functions f(x) and g(x), find (f ◦ g)(x)and (g ◦ f)(x). If these compositions are
______________________________________________________________________________________.
Examples:
Are f(x) and g(x) inverses of each other? Use composition to prove it algebraically.
1)
3)
𝑓(𝑥)=3𝑥−4
𝑔(𝑥)=
𝑥+4
3
𝑥3
8
3
𝑓(𝑥)=
𝑔(𝑥)= √8𝑥
2)
𝑓(𝑥)=4𝑥+8
𝑥
𝑔(𝑥)= −2
4
4)
𝑥−6
3
3
𝑔(𝑥)=
𝑥−6
𝑓(𝑥)=