Aim: How do we determine the inverse function if it exists?
Objective: to use the horizontal line test to determine if an inverse function exists and use algebra to find the
inverse function
Lesson Development: The idea of inverses, or opposites, is very important in mathematics. So important, in
fact, that the word is used in many different contexts, including the additive and multiplicative inverses of a
number. The actions of certain functions can be reversed as well. The rules governing the reversal themselves
can be functions.
f (5)  11
f (0)  3.5
g (11)  5
g (3.5)  0
g (5)  1
f ( g (5))  f (1)  5
ANS: (5, -3)
It would be 
g ( 1)  3
f ( g ( 1))  f ( 3)  1
(d) What is f 1 ( x ) ?
ANS: a) f 1 (2)  6, f 1 ( 4)  3
b) f 1 (0)  3
c)
d) It only takes two points to write the equation of a line: (2, 6) and (-4, -3)
Slope =
3
3
3
3
; y  6  ( x  2)  f 1 ( x )  x  3  6  x  3
2
2
2
2
OR We can solve for x in terms of y:
2
x  2  3y  2x  6
3
3y  6  2x
y
x
3y  6 3
3
 y  3  f 1 ( x )  x  3
2
2
2
EXISTENCE OF INVERSE FUNCTIONS
A function will have an inverse that is also a function if and only if it is one-to-one. Hence, a quick way to
know if a function has an inverse that is also a function is to apply the Horizontal Line Test. Also the inverse is
the reflection of the function across the line y = x. From geometry, remember for reflection across the line y =
x, we go from (a, b) to (b, a)?
HW#4: P466 – 467: 7 – 12, 19 – 20, 26*
*EC [+1 pt] Due when HW is due
Solutions
7) no
9) no
x3
11) f 1 ( x ) 
2
19) no
8) yes
12) f 1 ( x )  3x  6
20) yes; f 1 ( x)  3 ( x  2)
10) yes
Aim: How do we determine the inverse function if it exists?
(d) What is f 1 ( x ) ?
EXISTENCE OF INVERSE FUNCTIONS
A function will have an inverse that is also a function if and only if it
is one-to-one. Hence, a quick way to know if a function has an
inverse that is also a function is to apply the Horizontal Line Test.