Transcript No Slide Title

Chapter 5 Functions
5.3
MATHPOWERTM 11, WESTERN EDITION 5.3.1
Inverse Functions
The inverse of a function is a reflection of its graph in
the line y = x. It can be determined by interchanging
the coordinates of the ordered pairs in the functions.
The inverse of a function is written as f-1(x) and read as
“the inverse of f at x.”
When x and y are interchanged in the equation
of a function:
•The coordinates of the points that satisfy the
equation are interchanged.
•The graph of the function is reflected in the line y = x.
To determine the inverse of a function:
•Interchange x and y in the equation of the function.
•Solve the resulting equation for y.
5.3.2
Graphing the Inverse Function
Note: If the ordered pair (3, 6) satisfies
the function f(x), then the ordered pair (6, 3)
will satisfy the inverse, f-1(x).
Find the inverse of the function f(x) = 4x - 7.
Interchange the
x and y values.
y = 4x - 7
x = 4y - 7
x + 7 = 4y
x+7=y
4
x7
f (x) 
4
1
  7 
 0, 4  
(-3, 1)
(-7, 0)
 7  
 4 ,0 
(1, -3)
(0, -7)
5.3.3
Verifying an Inverse
If two functions f(x) and g(x) are inverses of
each other, then f(g(x)) must equal x and
g(f(x)) must equal x.
x 7
Verify that the functions f(x) = 4x - 7 and g(x) 
4
are inverses.
f(g(x)) must be equal to x.
x 7
g(x) 
4
 x  7    x  7  
f 
 4 
7




4
4
= (x + 7) - 7
=x
g(f(x)) must be equal to x.
f(x) = 4x - 7
4x  7  7

g(4x - 7) 
4x

4
=x
4
Since f(g(x)) and g(f(x)) are both equal to x, then f(x) and g(x)
are inverses of each other.
5.3.4
Graphing a Function and Its Inverse
Graph f(x) = x2 + 1
and its inverse.
(-2, 5)
(2, 5)
The graphs are
symmetrical about
the line y = x.
For the function:
Domain:
Range:
x R
y>1
For the inverse:
Domain: x > 1
y R
Range:
Is the inverse
a function?
(1, 2)
(-1, 2)
(5, 2)
(0, 1)
(2 , 1)
(1, 0)
(2, - 1)
(5, -2)
Could the domain
of f(x) be restricted
so that the inverse
is a function?
5.3.5
Graphing a Function and Its Inverse [cont’d]
Given f(x) = x2 + 1,
the inverse is
NOT a function.
Graph y = x2 + 1
where x > 0.
Graph the inverse.
Is this a function?
Graph y = x2 + 1
where x < 0.
Graph its inverse
Is this a function?
What are your
conclusions about
restricting the domain
so that the inverse is
a function?
Vertical Line Test
5.3.6
Questions:
Pages p. 99 #3, 5(a,c), 6(a,c,e), 10
and p. 105 #1(a,c), 2(a,c), 4, 6 - 10
5.3.7