Download 6.2 One-to-One and Inverse Functions

5-2: One-to-One Functions; Inverse Functions
If f(x) = y is a function, its inverse f –1(x) is f(y) = x, where y becomes the domain and x becomes the range. If
for each y in the domain of the inverse function there is a unique x in the range, it is a one-to-one function. If a
horizontal line intersects the graph of a function f no more than once, then f is one-to-one. Only one-to-one
functions have inverses. We can verify that f and f –1 are inverses showing that f ( f –1 (x)) = f -1 ( f (x)) = x.
The graph of a function f and its inverse f –1 are symmetric with respect to the line y = x. To find the inverse of
a function y = f(x), interchange x and y to obtain x = f(y) and solve for y in terms of x.
Find the inverse and determine whether the inverse is a function:
1. [(Bob, 68), (Dave, 92), (Carol, 87), (Elaine, 74), (Chuck, 87)].
2.
[(-2, 5), (-1, 3), (3, 7), (4, 12)
Use the horizontal line test to see whether f is one-to-one; if yes, graph its inverse:
(3)
(4)
(5) y = f(x) = -2x + 3
10
8
6
4
2
-10 -8
-6
-4
-2
2
4
6
8
10
-2
-4
-6
-8
-10
Verify that the functions f and g are inverses of each other:
1
6. f(x) = 3 – 2x; g(x) =  ( x  3)
7. f(x) = 2x – 7; g(x) = x  7
2
2
8. f(x) =
x2  5
3x  5 , g  x  
3
The following functions are one-to-one. Find the inverse and state the domain and range of each.
9.
2 x
3 x
#9
Domain
Range
f ( x) 
10.
f(x)
f -1(x)
f  x   x 2  4, x  0
#10
Domain
Range
f(x)
f -1(x)
TRY THESE
Which of the following graphs are one-to-one? Draw the graph of the inverse
function f-1 for those that are one-to-one.
1.
2.
3.
4.
Verify that the functions f and g are inverses of each other.
5. f  x    x  2 2 , x  2; g  x  x  2
6. f  x   x  5 ; g  x   3 x  5
2x  3
1  2x
Function f is one-to-one. Find its inverse; check your answer. State the domain of
f and find its range using f-1.
2
7. f  x   x  3 , x  0
3 x2
6-2: Exponential Functions
Exponential function is a function in the form f(x) = ax where a is a positive real number and a
 1. The domain of f is the set of all real numbers. The laws of exponents for real (irrational)
exponents are the same as those for integer and rational exponents. Irrational exponents are
1I
truncated to a finite number of digits, so that ar ax. The number e is the number that F
1 J
G
n
H nK
approaches as n.
f(x) = ax
Range
x-intercept
y-intercept
a>1
[0, )
None
(0, 1)
0<a<1
[0, )
None
(0, 1)
Horizontal
asymptote
x-axis as x 
x-axis as x

Characteristic
Increasing,
one-to-one
Decreasing
one-to-one
Passes
through
(0, 1) (1, a)
(0, 1)
(1, a)
Approximate to three decimal places:
1.
3
5
2.
2e
Graph the following exponential functions; state the domain, range, and any horizontal asymptote:
x
3. y = 2x
4. y = F
5. y = ex
G1 IJ = 2-x
H2 K
2x+2;
y=
y = 2x – 2
9.
y = -ex;
y = 2 - ex
y = 2-x-2;
y = 2-x + 2
Atmospheric pressure p measured in mm of mercury is related to the number of km h above sea level by
the formula p = 760e-0.145h. Find the pressure at a height of 2 km and at a height of 10 km.
Exponential Equations: If au = av, then u = v.
1 2 x
10. 5
1

5
11.  1 
 2
 
1 x
4
12.
 
e4
x
 e x  e12
2