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How can you determine
if you have a function?
• Vertical Line Test
• NO Duplicated “x” values
Graphing Inverse
Relations
Section 6-4
Pages 438-443
Objectives
• I can graph functions and their inverses
• I can use the Horizontal Line Test
Inverse Concept
• The main concept of an inverse is the x and
y coordinates have switched places
( x, y )
( y, x)
Inverses
• The inverse of any relation is obtained by
switching the coordinates in each ordered
pair of the relation.
• Example:
• { (1, 2), (3, -1), (5, 4)} is a relation
• { (2, 1), (-1, 3), (4, 5) is the inverse.
The ordered pairs of the function f are reversed to
produce the ordered pairs of the inverse relation.
Example: Given the function
f = {(1, 1), (2, 3), (3, 1), (4, 2)}, its domain is {1, 2, 3, 4}
and its range is {1, 2, 3}.
The inverse relation of f is {(1, 1), (3, 2), (1, 3), (2, 4)}.
The domain of the inverse relation is the range of the
original function.
The range of the inverse relation is the domain of the
original function.
GRAPHING AN
INVERSE FUNCTION
Given the graph of a relation, the graph of its
inverse is obtained by switching x- and ycoordinates.
The resulting graph is reflected about the line
y = x.
7
Graphing an Inverse
• Pick some Critical Points off
Original Graph (x, y)
• SWITCH the x and y values
• Re-plot the newly formed
ordered pairs.
9
GRAPH the inverse
The graphs of a relation and its inverse are reflections
in the line y = x.
Example: Find the graph of the inverse relation
3
(
x
 2)
geometrically from the graph of f (x) =
4
y
The ordered pairs of f are 3
( x  2)
given by the equation y 
.
4
The ordered pairs of the inverse are
3
(
y
 2) .
given by x 
4
y=x
2
x
-2
( y  2)
x
4
3
2
-2
( x 3  2)
y
4
Horizontal Line Test
A function y = f(x) is one-to-one if and only if
no horizontal line intersects the graph of y = f(x)
in more than one point.
Example: The function
y = x2 – 4x + 7 is not one-to-one
on the real numbers because the
line y = 7 intersects the graph at
both (0, 7) and (4, 7).
y
(4, 7)
(0, 7)
y=7
2
x
2
Example: Apply the horizontal line test to the graphs
below to determine if the functions are one-to-one.
b) y = x3 + 3x2 – x – 1
a) y = x3
y
-4
y
8
8
4
4
4
-4
4
x
one-to-one
x
not one-to-one
Home Work
• Worksheet 10-1