Algebra 2 Notes AII.7 Functions: Composite, Inverse
Mrs. Grieser
Name: _______________________________ Date: _____________ Block: _______
Composite Functions
Given: f(x) = x2 + 1
1) Find…
f(2) = _____________
f(0) = _____________
f(-3) = ______________
f(x2) = _____________
f(x + 1) = _____________
2) Find…
f(y) = _____________
3) If g(x) = 2x, find f(g(x)) _______________________________________________________
A composite function is a function where the range (output) of one function is the
domain (input) of another
We write a composition of functions as: f(g(x)) or (f◦g)(x)
Example 1: Given f(x) = x + 1 and g(x) = -2x,
find g(f(3)).
Example 2: f(x) = x + 1 and g(x) = 2x. Find:
(f◦g)(x) __________________
f(g(1)) ____________
f(g(2)) ______________ f(g(3))______________
Example 3: Given the functions f(x) = 5x and g(x) = x2, find:
a) (f◦g)(x)____ b) (g◦f)(x)______
Question: Is it true that (f◦g)(x) = (g◦f)(x)?______ Can you think of a case where it is?_______
Example 4: Given the functions p(x) = x + 2 and h(x) = x2, find: a) (h◦p)(3) ___ b) (h◦p)(x) ____
You Try…
1) What are the domain and range of h(g(x)) at
right?
2) Given g(x) = x2 and h(x) = x - 3, draw an input-
output diagram to illustrate (g◦h)(-3). What is
the output?
3) Given f(x) =
a) f(g(x))
x , g(x) = (x+1)2, and h(x) = -2x, find
b) (h◦f)(8)
c) g(h(m))
d) (f◦h)(-2)
1
e) (g◦f◦h)(-8)
Algebra 2 Notes AII.7 Functions: Composite, Inverse
Mrs. Grieser
Inverse Functions
What does the word “inverse” mean?____________________________________________
An inverse function “undoes” another function.
Some inverses we know: sin(x) and sin-1(x) ; cos(x) and cos-1(x); tan(x) and tan-1(x)
o Explain what the inverse does here____________________________________________
If the following diagram illustrates a function, what would the inverse be?
Inverse:
Given function {(−3, 27) (−2, −8) (−1, −1) (0, 0) (1, 1) (2, 8) (3, -27)}, what is the inverse?
__________________________________________________________________________________
o Is the inverse relation a function?_____________________
Given function {(−3, 9) (−2, 4) (−1, 1) (0, 0) (1, 1) (2, 4) (3, 9)}, what is the inverse?
__________________________________________________________________________________
o Is the inverse relation a function?_____________________
Is the inverse of a function always a function? ___________
One-to-One Functions
When the inverse of a function is a function, too, then we call the function one-to-one.
o A function is one-to-one IFF its inverse is a function.
o In one-to-one functions, not only does every input have OAOO output, but every
output has OAOO input! (OAOO = one and only one)
o Vertical line tests tell us if a relation is a function; horizontal line tests tell us if a
function is one-to-one
o Which of the following functions are one-to-one? Which of the following functions
will have inverses that are also functions?
a)
b)
c)
one-toone?__________
one-toone?_________
inverse is a
function?_____
inverse is a
function?_____
2
one-to-one?______________
inverse is a function?_______
Algebra 2 Notes AII.7 Functions: Composite, Inverse
Mrs. Grieser
Finding the Inverse of a Function
Suppose f(x) = 2x. What is its inverse?
o Look at a sample of the function mapping:
2→4,
3→6,
4→8,
5→10
o The inverse mapping would do the opposite:
4→2,
6→3,
8→4,
10→5
An inverse “undoes” a function, so do the opposite operation.
1
In this case, divide by 2. The inverse of f(x) = 2x is f-1(x)= x
2
What is the inverse of f(x) = 3x – 1?
o Do the opposite: add 1, then divide by 3; f-1(x) =
x 1
3
Method for finding inverses of functions:
Switch y and x
Solve for y
Example: f(x) = 2x + 3. Find f-1(x).
o Re-write using y: y = 2x + 3
o Switch x and y:
x = 2y + 3
o Solve for y:
y=
o Conclusion:
f-1(x) =
x 3
2
x 3
2
Relationship Between Composite Functions and Inverses
g(f(x)) = f(g(x)) = x IFF f and g are inverses.
Example: f(x) = 3x; g(x) =
1
x; use composition of functions to show that f and g are
3
inverses.
o Find f(g(x)) and g(f(x))_________________________________________
You try… verify whether the pairs of functions are inverses of each other:
a) h(x) = x3, g(x) =
3
x
b) h(g(x))=x; g(h(x))=x
3
c) f(x) =
1
x 1 ; g(x) = 5x + 5
5
Algebra 2 Notes AII.7 Functions: Composite, Inverse
Mrs. Grieser
Graphs of Inverses
Find the inverse of the function, then graph the function and its inverse.
f(x) = x2
Find inverse:
Graph:
Graphs of inverses ____________________. They reflect over the line ________________.
Restricting Domain and/or Range
Sometimes we restrict the domain and/or range so that the inverse is a function. In the
above example, how could we restrict domain and/or range to make the inverse a function?
You Try…
Find the inverses of the functions. Are the inverses one-to-one? Are the inverses
functions? Use graphs to help you decide.
1) f(x) = -2x + 5
2) f(x) =
3) f(x) = 2x2 + 1
2
x 2
3
one to one?_______
one to one?_______
one to one?_______
inverse is a functon?______
inverse is a functon?______
inverse is a functon?______
4) f(x) = x3
5) f(x) = x7
one to one?_______
one to one?_______
inverse is a functon?______
inverse is a functon?______
Verify that f and g below are inverses.
6) f(x) = x + 4; g(x) = x - 4
7) 7) f(x) = 5x2 - 2; g(x) =
x 2
5
1
2
8) True or false: if f(x) = xn and n is an even positive integer, then f-1 is a function.
9) True or false: if f(x) = xn and n is an odd positive integer, then f-1 is a function.
10) Write a function f such that f-1 is a line with slope 3.
4