Algebra 2/Trig
Name: ____________________________________
Section 2.5 Notes: Inverse Functions p. 118
Objectives: Find the inverse of a relation or function. Determine whether the inverse of a function is a function.
INVERSE OF A RELATION
The inverse of a relation consisting of the ordered pairs (x, y), is the set of all ordered pairs __________.
The domain of the inverse is the ______________________ of the original relation.
The range of the inverse is the ________________________ of the original relation.
An inverse relation _____________________________ input and output values of the original relation. This means
the domain and range are also interchanged.
Example 1: Find the inverse of each relation. Then, state whether the relation is a function and if its inverse is a
function.
a. (1, 2), (2, 4), (5, 6), (4, 9)
Relation:
Inverse:
The given relation _____ a function because each ____________ value is paired with exactly one _______________
value. The inverse ______________ a function because each _______ value is paired with exactly one
_______________ value.
b. (1, 4), (1, 6), (3, 6), (4, 9)
Relation:
Inverse:
The relation ______________ a function because the domain value of 1 is paired with ______ range values,
______________. The inverse _____________ a function because the domain value 6 is paired with
______________ values. ________________________.
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Example 2: Graph y = 2x -1. Sketch it on the graph to the right.
a. Make a table for y = 2x – 1
x
2
1
0
-1
-2
y
b. Find the inverse relation for y = 2x – 1.
Step 1: Switch x and y
Step 2: Solve for y. This is the inverse relation. Using function notation: 𝑓 −1
c. Graph the inverse on the same grid (above).
d. Complete the table for the inverse.
x
y
e. How do the tables for y = 2x -1 and y = ½ x + ½ compare?
f.
Graph y1 = 2x – 1and y2 = (its inverse) on your calculator. Then graph y3 = x.
Use Zoom and Z-Standard (6) to get a square window.
Notice that the graphs of y1 and y2 are symmetric about the line y = x.
INVERSE FUNCTIONS
Functions f and g are inverses of each other if:
f o g = x and g o f = x
VERIFY FUNCTIONS ARE INVERSEES
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Example 3: Verify that 𝑓(𝑥) = 3𝑥 − 5 and 𝑓(𝑥) = 3 𝑥 + 3 are inverse functions.
Step 1: Show that f o f-1 = x
Step 2: Show that f-1 o f = x
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FIND THE INVERSE OF A POWER FUNCTION
Example 4: f(x) = 64x3
Step 1: Replace 𝑓(𝑥) with 𝑦
Step 2: Switch x and y
Step 3: Solve for y.
INVERSE OF A REAL-LIFE PROBLEM
When you are finding the inverse of a model, do NOT switch the variables. Just solve for the other variable.
Switching the variables would be confusing because they represent real-life quantities.
Example 5: If the cost of repairing a car has the equation: c = 25h + 50
To find the inverse, solve for h.
THE HORIZONTAL LINE TEST
The inverse of a function f is also a function if and only if no horizontal line intersects the graph of f more than once.
Inverse is a Function
Inverse is NOT a function
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Homework: Pages 122 & 123. Problems 12 – 48 Even and 51 & 52 (42 – 48 you may want to use a graphing
calculator – do these first if you do not have one at home, or graph by hand)
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