Chapter 5: Exponential and Logarithmic Functions. 5.1 Inverse Functions 5.2 Exponential Functions 5.3 Logarithms and Their Properties 5.4 Logarithmic Functions 5.5 Exponential and Logarithmic Equations and Inequalities
5.1 Inverse Functions
5.2 Exponential Functions
5.3 Logarithms and Their Properties
5.4 Logarithmic Functions
5.5 Exponential and Logarithmic Equations and Inequalities
5.6 Further Applications and Modeling with Exponential and Logarithmic Functions
Also, f[g(12)] = 12. For these functions, it can be
for any value of x. These functions are inverse functions
of each other.
A function f is a one-to-onefunction if, for elements a and b from the domain of f,
a b implies f(a) f(b).
Example Decide whether the function is one-to-one.
(a) For this function, two different x-values produce two different y-values.
(b) If we choose a = 3 and b = –3, then 3 –3, but
If every horizontal line intersects the graph of a function at no more than one point, then the function is one-to-one.
Example Use the horizontal line test to determine
whether the graphs are graphs of one-to-one functions.
Let f be a one-to-one function. Then, g is the inversefunction of f and f is the inverse of g if
are inverse functions of each other.
Finding the Equation of the Inverse of y = f(x)
1. Interchange x and y.
2. Solve for y.
3. Replace y with f-1(x).
Any restrictions on x and y should be considered.
Example Find the inverse, if it exists, of
Write f(x) = y.
Interchange x and y.
Solve for y.
Replace y with f-1(x).
If a function is one-to-one, the graph of its inverse f -1(x) is a reflection of the graph of f across the line y = x.
Solution Notice that the domain of f is restricted
to [–5,), and its range is [0, ). It is one-to-one and
thus has an inverse.
The range of f is the domain of f-1, so its inverse is
Example Use the one-to-one function f(x) = 3x + 1 and the
numerical values in the table to code the message BE VERY CAREFUL.
A 1 F 6 K 11 P 16 U 21
B 2 G 7 L 12 Q 17 V 22
C 3 H 8 M 13 R 18 W 23
D 4 I 9 N 14 S 19 X 24
E 5 J 10 O 15 T 20 Y 25
Solution BE VERY CAREFUL would be encoded as
7 16 67 16 55 76 10 4 55 16 19 64 37
because B corresponds to 2, and f(2) = 3(2) + 1 = 7,
and so on.