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Chapter 4
College Algebra
4.1 Inverse Functions
 Inverse Relations
 Interchanging the first and second coordinates of
each ordered pair in a relation produces the inverse
relation.
 G = {(2,4), (-1,3), (-2,0)}
 The inverse relation is : {(4,2), (3,-1), (0,-2)}
 If a relation is defined by an equation, interchanging
the variables produces an equation of the inverse
relation.
 y = x2 – 5x
 The inverse relation: x = y2 – 5y
One-to-One Function
 A function f is one-to-one if different inputs have
different outputs – that is
if a ≠ b, then f(a) ≠ f(b).
 Or a function f is one-to-one if when the outputs
are the same, the inputs are the same – that is
if f(a) = f(b) then a = b
 Horizontal-Line Test
 If it is possible for a horizontal line to intersect the
graph of a function more than once, then the
function is not one-to-one and its inverse is not a
function.
Properties of One-to-One
Functions and Inverses
 If a function f is one-to-one, then its inverse f-1 is a
function.
 The domain of a one-to-one function f is the
range of the inverse f-1.
 The range of a one-to-one function f is the
domain of the inverse f-1.
 A function that is increasing over its domain or is
decreasing over its domain is a one-to-one
function.
Obtaining a Formula for an
Inverse
 If a function f is one-to-one, a formula for its
inverse can generally be found as follows:
 Replace f(x) with y, when necessary.
 Interchange x and y.
 Solve for y.
 Replace y with f-1(x).
 The graph of f-1 is a reflection of the graph of f
across the line y = x.
 If a function f is one-to-one, then f-1 is the unique
function such that each of the following holds:
( f -1 f )(x) = f -1 ( f (x)) = x
( ff -1 )(x) = f ( f -1 (x)) = x
4.2 Exponential Functions
and Graphs
 Exponential Functions
 The function f(x) = ax, where x is a real number, a > 0
and a ≠ 1, is called an exponential function, base a.
 Properties
 Continuous
 One-to-One
 Domain: (-∞,∞)
 Range: (0,∞)
 Increasing if a > 1
 Horizontal asymptote: y = 0
 y-intercept (0,1)
The Number e
 e = 2.7182818284…
 The graph of y = ex
 The inverse of the exponential
graph is the graph of the natural
log, ln.
Homework
4.3 Logarithmic Functions
and Graphs
 Logarithmic Function, Base a
 We define y = logax as that number y such that ay=x
where x > 0 and a is a positive constant other than 1.
 Properties
 loga1 = 0 and logaa = 1 (for any log base a)
 logax = y  x = ay
 Logarithmic Function, Base 10
 “log10x” read “the logarithm, base 10 of x” means
“the power to which we raise 10 to get x”
 log base 10 is called the common logarithm
 This is the base that calculators use
Natural Logarithms
 Logarithms with the base of e are called natural
logarithms.
 The abbreviation is ln
 ln x means logex
 ln 1 = 0 and ln e = 1, for the logarithmic base e.
 Change of base formula
 For any logarithmic bases a and b, and any positive
number M,
a
b
a
 Why might this be important? Could we use ln?
log M
log M
log b
4.4 Properties of Logarithmic
Functions
 The Product Rule
 For any positive numbers M and N and any
logarithmic base a,
logaMN = logaM + logaN
 The Power Rule
 For any positive number M, any logarithmic base a,
and any real number p,
logaMp = p logaM
 The Quotient Rule
 For any positive numbers M and N and any
logarithmic base a,
logaM/N = logaM - logaN
Simplifying Expressions
 The Logarithm of a Base to a Power
 For any base a and any real number x,
logaax = x
 A Base to a Logarithmic Power
 For any base a and any positive real number x,
alogax = x
Homework
4.5 Solving Exponential and
Logarithmic Equations
 Equations with variables in the exponents are
called exponential equations.
 Base-Exponent Property
 For any a > 0, a ≠ 1, ax = ay > x = y
 Property of Logarithmic Equality
 For any M > 0, N > 0, a > 0, and a ≠ 1,
logaM = logaN 
M=N
 Equations containing variables in logarithmic
expressions are called logarithmic equations.
4.6 Applications and Models
 Exponential Growth Rate
 The function P(t) = P0ekt k > 0
is a model of many kinds of population growth
whether it be a population of people, bacteria, cell
phones, or money. In this function, P0 is the
population at time 0, P is the population after time t,
and k is called the exponential growth rate.
 Interest Compounded Continuously
 Suppose that an amount P0 is invested in a savings
account at interest rate k compounded
continuously. The amount P(t) in the account after t
years is given by the exponential function
P(t) = P0ekt
4.6 Applications
 Growth Rate and Doubling Time
 The growth rate k and the doubling time T are related
by:
 kT = ln 2
 k = (ln 2) / T
 T = (ln 2) / k
 Exponential Decay
 The function P(t) = P0 e-kt k > 0, is an effective model of
the decline or decay of a population. In this case, P0 is
the amount of the substance at time 0, and P(t) is the
amount of the substance after time t, where k is a
positive constant that depends on the situation and is
called the decay rate.
Models
 Model for Limited Growth
 The Logistic Function
 P(t) = a / (1 + be-kt)
 This function increases towards a limiting value a
as t approaches infinity.
 Another model of limited growth is provided by
 P(t) = L(1 – e-kt), k > 0
Homework