# College Algebra

**13 Inverse Functions**

**Definition 13.1. **Two functions f and g are said to be** inverse
functions**

iff the graphs the equations y = f(x) and x = g(y) are the same, i.e. iff

y = f(x) x = g(y).

We also say that g is the** inverse** of f and write

g = f^{ -1}.

**Example 13.2. **Since

the functions f(x) = 3x + 7 and g(y) = (y - 7)/3 are inverse functions.

**Remark 13.3**. Be careful not to confuse f(x)^{-1} and f^{ -1}(x).
For example,

if f(x) = 3x + 7, then

**13.4.** If f and g are inverse functions, then the range of
f is the domain of g

and the domain of f is the range of g. To decide if a function has an inverse

we apply the

Horizontal Line Test. A function y = f(x) has an
inverse if andonly if every horizontal line y = b intersects the graph in at most one point. Then the horizontal line y = b intersects the graph y = f(x) in the point P(f ^{ -1}(b), b). |

**13.5. **A function f is said to be **one-to-one **iff

Saying that a function is one-to-one is just another way of saying its graph

satisfies the horizontal line test, i.e. that the function has an inverse. The

function f(x) = x^{2} is not one-to-one since f(3) = f(-3) but 3 ≠ -3.

**Definition 13.6.** A function f is said to be** increasing** iff

A function f is said to be **decreasing** iff

**
Theorem 13.7.** If a function either increasing or decreasing it is one-to-one

and therefore has an inverse.

Proof. Assume f is increasing and that . Then it is not the

case that since that would imply and it is not the case

that since that would imply . The only possibility us

that .

**Remark 13.8.** Just because a function has an inverse
doesn't mean that we

can find a formula for the inverse. For example, the function f(x) = x^{5} + x

is increasing and therefore has an inverse f^{ -1}, but in graduate courses in

algebra it is proved that there is no elementary formula for f^{ -1}(y), i.e. there

is no expression for f^{ -1}(y) involving only the operations we have have defined

in these notes. Put another way, there is no nice formula for the solution of

the equation 3 = x^{5} + x.

Even if we can't find a formula for the inverse of a function, we can still

compute its value for any given input to any degree of accuracy with (say)

a computer. So we can give the inverse a name and compute with it using

the rules of algebra. This is exactly what we shall do in Section** 17**. with the

exponential function

f(x) = 2^{x}.

This function is increasing so has an inverse function. and the inverse func-

tion is denoted by :

**13.9.** In Section **12.3** we said that, unless the contrary is stated, the domain

of a function f(x) which is defined by an expression in x is the set of x for

which the expression is meaningful. However, we may decide to restrict the

domain to a smaller set in order to define an inverse function. For example,

in Section **13.5** we saw that the function f(x) = x^{2} is not one-to-one since

f(3) = f(-3) and hence this function does not have an inverse function. But

we can define

q(x) = x^{2}. x ≥ 0

so the domain is [0,∞). Then

so q(x) and are inverse functions. (This device is used in trigonometry to

define the inverse trigonometric functions.) If we were to program a computer

to compute the function q(x), asking the computer to compute q(-3) would

produce an error message like \function undefined for this input".

**13.10. **If the function f has an inverse, then the graph of the equation

y = f(x) is the same as the graph of the equation x = f^{ -1}(y). Of course

the graph of the equation y = f^{ -1}(x) is (usually) different. The graphs of

the equations y = f(x) and y = f^{ -1}(x) are obtained from
each other by

interchanging the x-axis and the y-axis, i.e. by reflecting in the line y = x.

**14 Average Rate of Change
**

**Definition 14.1.**The

**average rate of change**of the function f over the

interval [a, b] is the slope of the line joining the point (a, f(a)) to the point

(b, f(b)). When y = f(x) the average rate of change is often written as

i.e. the average rate of change is the change Δy = f(b) -
f(a) in y divided

by the change Δx = b - a in x.

**Example 14.2. **The average rate of change of the function f(x) = x^{2} on

the interval [a, b] is

Note that the b-a in the denominator cancels out. To see
way this happens

we will repeat the calculation using the notation h = b - a = Δx so that

b = a + h. When we expand the numerator the terms which don't contain

an h cancel. Then the h in the denominator cancels with (some of) the h's

in the numerator as follows:

This cancellation will always happen when you are asked to
simplify an

average rate of change.

**14.3.** The average rate of change of a function f over the interval [a, b] is

undefined when a = b (zero divided by zero is nonsense) but for b very close

to a the average rate of change will usually be close to a number called the

**instantaneous rate of change.** This is the slope of the tangent line to the

graph y = f(x) at the point (a, f(a)).

In calculus the instantaneous rate of change is called the
**derivative**. In the

previous paragraph we saw that for the function f(x) = x^{2} the average rate

of change over the interval [a, b] is a+b. This is 2a+h where h =Δx = b-a.

The instantaneous rate of change at the point (a, f(a)) is 2a which is obtained

from the average rate of change by taking h = 0.

**14.4.** Suppose you are traveling from Madison to Milwaukee by automobile

via the interstate highway I-94. Your position is a function s = f(t) of the

time t. The value s is the number on the mile marker at the side of the

road. (Along much of the road there is a mile marker every tenth of a mile,

but imagine there is one every few feet.) The change Δs = f(t + Δt) - f(t)

is also the change in the odometer reading in your car. If the time interval

Δt is so short that your speed doesn't change much in the time interval, the

average rate of change Δs/ Δt is your speed as shown on the speedometer.

The speed is the instantaneous rate of change of the position.

**15 Polynomials**

**15.1.** For any graph the points where it intersects the x-axis are called the

x-intercepts and the points where it intersects the y-axis are called the y-

**intercepts**. The equation of the x-axis is y = 0 so we find
the x-intercepts

of the graph of an equation by plugging in y = 0 and solving for x. Similarly

the equation of the y-axis is x = 0 so we find the y-intercepts by plugging in

x = 0 and solving for y. If the graph is the graph of a function y = f(x),

then (assuming that 0 is in the domain) the y-intercept is the point (0, f(0)),

and the x-intercepts are the points (r, 0) such that f(r) = 0. The numbers

r such that f(r) = 0 are called the** zeros** (and sometimes also the **roots**) of

the function.

**Definition 15.2. A polynomial **is a function of form

where the exponents on x are nonnegative integers. A **polynomial equation**

is an equation of form

Its solutions are the zeros of the polynomial and are also called the** roots** of

the polynomial. When ≠ 0 we say that the** degree
**of the polynomial is n.

The constants are called the coefficients. For any polynomial

f(0) = (the **constant term**) so the y-intercept of the graph is the point

**15.3. **A polynomial of degree one (or zero) is called** linear** (since its graph

is a line), a polynomial of degree two is called **quadratic**, and a polynomial

of degree three is called **cubic**. In Paragraph **6.4** we used the notation

f(x) = mx + b (slope-intercept form) rather than
for a

linear function and in Theorem **11.2** we wrote the quadratic equation as

ax^{2} + bx + c = 0 rather than as .

**15.4.** It is easy to graph a linear function: just find two points

and on the graph and draw the line through them. It is also not

hard to graph a quadratic function f(x) = ax^{2} + bx + c. The graph is a

parabola which opens up (like y = x^{2}) if a > 0 and down (like y = -x^{2})

if a < 0. By the Quadratic Formula in Theorem **11.2** the graph has two

x-intercepts if the **discriminant** b^{2} - 4ac is positive, one x-intercept if the

discriminant is zero, and no x-intercept if the discriminant is negative.

**Remark 15.5.** When a = 1 and b and c are integers we could
try to solve

the quadratic equation x^{2} + bx + c = 0 by factoring. If

(for all x) then

If we suspect that the roots are integers, we can try all possible ways of

factoring c. For example, to solve x^{2} - 5x + 6 = 0 we try
,

(-1,-6), (2, 3), (-2,-3). Only gives
so

x^{2} - 5x + 6 = (x - 2)(x - 3)

and the solutions of x^{2} - 5x + 6 = 0 are x = 2 and x = 3. Of course, there

is usually no reason to suspect that the roots are integers so it is best to use

the Quadratic Formula (Theorem **11.2**) to solve a quadratic equation.

**15.6. **A polynomial inequality like

is easy to solve if we can factor the left hand side. Then we can write it in

the form

On each of the intervals the sign
of

the polynomial is constant so we can compute the sign by evaluating the

polynomial at some point in the interval. For example,

(x - 1)(x - 4)(x - 9) > 0
x in (1, 4) ∪ [ (9,∞)

and

(x - 1)(x - 4)(x - 9) ≥ 0 x in [1, 4] ∪ [ [9,∞)

As a check we evaluate

at x = 0 in (-∞, 1) and get (0 - 1)(0 - 4)(0 - 9) = -36 < 0,

at x = 2 in (1, 4) and get (2 - 1)(2 - 4)(2 - 9) = 14 > 0,

at x = 5 in (4, 9) and get (5 - 1)(5 - 4)(5 - 9) = -16 < 0, and

at x = 10 in (9,∞) and get (10 - 1)(10 - 4)(10 - 9) = 54 > 0.

**15.7.** To graph a polynomial f(x) we use the following rules:

(i) The sign of the polynomial does not change between two
adjacent roots.

To determine this sign we can evaluate the polynomial at any number

in between.

(ii) If the polynomial can be factored so that

where are the roots, then between each pair of

roots the graph reverses direction exactly once, i.e. either the

function value f(x) increases on some interval
and then decreases

on the interval or it decreases on some
interval and then

increases on the interval

(iii) The absolute value of f(x) is large when the absolute value of x is large.

Whether f(x) is large positive or large negative depends only on the

sign of the coefficient of x^{n} (where n is the degree) and on whether n

is odd or even.

In calculus you will learn how to determine on which intervals the function

is increasing and you will learn enough to understand why (ii) is true.

Below we have graphed the polynomial

f(x) = x^{4} - 2x^{3} - x^{2} + 2x = (x + 1)(x - 0)(x - 1)(x - 2)

which factors completely as in item (ii). The degree is 4 and the roots are

-1, 0, 1, 2. In each of the intervals (-1, 0), (0, 1), (1, 2) between roots the

function reverses direction exactly once. Next to the graph y = f(x) is the

graph y = f(x)-1. The degree is still 4 but the graph y = f(x)-1 has two

x-intercepts (not 4 as does y = f(x)) and between them the function reverses

direction three times. According to item (ii) this means that the polynomial

f(x) - 1 does not factor completely (into real linear polynomials).