Today in Pre-Calculus
• Go over homework questions
• Notes: Inverse functions
• Homework
Inverse Functions
• Reversing the x- and y-coordinates of all the ordered
pairs in a relation gives the inverse.
• The inverse of a relation is a function if it passes the
horizontal line test.
• A graph that passes both the horizontal and vertical
line tests is a one-to-one function. This is because
every x is paired with a unique y and every y is paired
with a unique x.
Inverse Functions
• Definition: If f is a one-to-one function with domain D
and range R, then the inverse function of f,
denoted f –1, is the function with domain R and range
D defined by f –1(b)=a iff f(a)=b
Graphing Inverses
Example
a) f(x) = 2x – 3
y = 2x – 3
x = 2y – 3
x + 3 = 2y
x : (-∞,∞), y: (-∞,∞)
y : (-∞,∞), x: (-∞,∞)
1
3
y  x
2
2
1
3
f ( x)  x 
D: (-∞,∞)
2
2
-1
Example
f(x) = x
y= x
x = [0,∞) , y = [0,∞)
x= y
y = [0,∞) , x = [0,∞)
y = x2
f –1(x) = x2 D=[0,∞)
Example
x
f ( x) 
x2
x
y
x ≠ -2 , y ≠ 1
x2
y
x
y ≠-2 , x ≠1
y2
x(y+2) = y
xy + 2x = y
2x = y – xy
2x = y(1-x)
2x
y
1 x
2x
f ( x) 
1 x
1
D : (,1)  (1, )
Inverse Composition Rule
states that a function f is one-to-one with inverse function g iff
f(g(x)) = x for every x in the domain of g and g(f(x)) = x for
every x in the domain of f.
Used to verify that f and g are inverses of each other.
Example
x3
f ( x)  2 x - 3 and g ( x) 
2
 x3
 x3
f ( g ( x))  f 
  2
 3  x 33  x
 2 
 2 
(2 x  3)  3 2 x
g ( f ( x))  g  2 x  3 

x
2
2
Example
f ( x)  x and g ( x)  x 2
 
f ( g ( x))  f x
g ( f ( x))  g
2
x x
 x  x
2
2
x
Example
x
2x
f ( x) 
and g ( x) 
x2
1- x
2x
2x
2x
2x
 2x 
1 x 
f ( g ( x))  f 



x

2x
 1 x 
 2 2 x  2(1  x) 2 x  2  2 x 2
1 x
x
2
x
2x
2x
2x


x

2
g ( f ( x))  g 



x

1( x  2)  x x  2  x 2
 x  2  1 x
x2
Homework
• pg 135: 13 – 31 odd
• Quiz: Tuesday, October 8
• Chapter 1 test: Friday, October 11

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