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Functions

Announcements

• Homework 2 due this Friday in class.
– Please write your Section number on your homework.
– Staple pages together.

• Two volunteer note takers needed.

Recall:

•A function f is a rule that assigns to each element x of a set
X a unique element y of a set Y .

f (x) = y

•The element y is called the image of x under f and is denoted
f (x). The set X is called the domain of f, and the set of all
images of elements of X is called the range of the function.

•If the range of f consists of all of Y , then f is said to map
X onto Y . If each element in the range is the image of one
and only one element in the domain, then f is said to be a
one-to-one function.

•A function f is bounded on [a, b] if there exists a number B
such that |f (x)| ≤B for all x in [a, b].

•y is called the dependent variable and x is called the
independent variable.

Domain Convention

• Domain of f is assumed to be the set of real numbers for which
the function is defined (unless otherwise specified).

• Examples: find the domain:

Function equality

• Two functions f and g are equal if and only if
1. f and g have the same domain
2. f (x) = g (x) for all x in the domain

• Example:

Function composition

The composite function f o g is defined by

for each x in the domain of g for which g (x) is in the domain of f.

• Example:

We can also express a given function as a composition of two
functions.

• Example:

Example application

Suppose a study showed that when the population is p hundred
thousand people, the average daily level of carbon monoxide in the
air is given by

parts per million (ppm).

A second study showed that t years from now, the population will
be

hundred thousand people.

What will the level of air pollution be in t years?

Solution:

• The graph of a function consists of points whose coordinates
(x, y) satisfy y = f (x), for all x in the domain of f.

• Vertical line test for function

• Intercepts of a graph: y-intercept is the point (0, b) where
b = f(0). An x-intercept is a point (a, 0) where f (a) = 0.

• Note that there can be at most one y-intercept. There may be
more than one x-intercept.

• To find x-intercepts, solve f (x) = 0 for x.

• To find y-intercept, calculate f(0) (if 0 is in the domain).

Symmetry

• symmetric with respect to the y-axis: f(−x) = f (x)
(even function)
Examples: f (x) = cos x, f (x) = x²
• symmetric with respect to the origin: f(−x) = −f (x)
(odd function)
Examples: f (x) = sin x, f (x) = x³
• Some functions are even, some are odd, some are neither.

• See sample graphs in Table 1.3.

Classification of Functions

Algebraic functions:

• Polynomial:

• a_n is the leading coefficient

• a₀ is the constant term

• If a_n ≠ 0, n is the degree of the polynomial.
E.g., a constant function is zero degree,
a linear function is first degree,
a quadratic function is second degree,
etc.

• A rational function is the quotient of two polynomial
functions, p (x) and q (x),

• Power functions: f (x) = x^r (r nonzero, real)
– integer powers: (pos. integer n)
– reciprocal powers:
– roots:

Transcendental functions:

• Trigonometic functions: f (x) = cos x

• Exponential functions: f (x) = b^x

• Logarithmic functions:

Section 1.4: inverse functions

•

• If f has an inverse f^-1, it is the function that reverses the
effect of f:

• If the inverse of a function is itself a function, we have the
following definition:

Let f be a function with domain D and range R. Then the
function f^-1 with domain R and range D is the inverse of f if

for all x in D
and

for all y in R

• If f has an inverse, the inverse is unique.

• Not every function has an inverse. (E.g., f (x) = x².)

• A function f will have an inverse f^-1 on the interval I when
there is exactly one number in the domain associated with each
number in the range.

• I.e., a function has an inverse only if it is one-to-one.

• “Horizontal line test”

• A function is strictly increasing on an interval I if its graph
is always rising on I

implies

• A function is strictly decreasing on an interval I if its graph
always falls on I

implies

• A function is strictly monotonic on an interval I if it is
either strictly increasing or strictly decreasing throughout that
interval.

Theorem: A strictly monotonic function has an inverse.
Lef f be a a function that is strictly monotonic on I. Then f^-1
exists and is strictly monotonic on I (strictly increasing if f is
strictly increasing and strictly decreasing if f is strictly decreasing).

Graphing inverse functions

• If (a, b) is a point on the graph of f, then (b, a) is a point on
the graph of f^-1.

• You can graph f^-1 by reflecting the graph of f about the line
x = y

Inverse Trig Functions

• The trigonometric functions are not one-to-one, so they do not
in general have inverses.

• But we can restrict them to intervals on which they are
one-to-one and do have inverses.

• E.g., sin (x) is strictly increasing on