• Homework 2 due this Friday in class.
– Please write your Section number on your homework.
– Staple pages together.
• Two volunteer note takers needed.
•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
•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
• 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:
• 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
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.
We can also express a given function as a
composition of two
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).
hundred thousand people.
|• The graph of a function consists of points
(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).
• symmetric with respect to the y-axis: f(−x) = f (x)
Examples: f (x) = cos x, f (x) = x2
• symmetric with respect to the origin: f(−x) = −f (x)
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
• an is the leading coefficient
|• A rational function is the quotient of two
functions, p (x) and q (x),
• Power functions: f (x) = xr (r nonzero, real)
• 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
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
for all y in R
• If f has an inverse, the inverse is unique.
|• Not every function has an inverse. (E.g., f (x)
• 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
• A function is strictly decreasing on an interval I if its graph
always falls on I
• A function is strictly monotonic on an interval I if it is
either strictly increasing or strictly decreasing throughout that
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 . If we restrict
sin (x) to that interval, it does have an inverse, sin -1.
if and only if x = sin (y) and
• Similar process for the other trigonometric functions. See Table
1.4 in Strauss.
• for all x