Math 31A Discussion Notes
Week 0
September 24, 2015
Since lecture hasn’t met yet, this week’s discussion topics won’t follow the most logical
outline. Instead, we have a smattering of pre-calculus topics that you should feel quite
comfortable with coming into the term. If any of these topics are new or challenging to you,
you should seek out help early in order to catch up. Khan Academy is a great place to brush
up your pre-calculus skills.
Functions
Definition. A function is a correspondence between two sets (called the domain and
range, respectively) which is consistent, meaning that each value in the first set corresponds
to exactly one element in the second set.
Example.
1. The correspondence {(2, 7), (3, 12), (5, 28), (−1, 4), (0, 3), (1, 4)} (where (a, b) means “a
corresponds to b”) is a function between the sets {−1, 0, 1, 2, 3, 5} and {3, 4, 7, 12, 28}.
2. The correspondence {(2, 7), (3, 12), (5, 28), (5, 4), (0, 3), (1, 4)} is not a function between
the same sets. Why?
3. Given that x2 −y = 4, can we determine y as a function of x? What about determining
x as a function of y?
√
4. Some functions with which you should be familiar include sin, cos, tan, f (x) = + x,
g(x) = |x|, h(x) = xn . What are the domains and ranges of these functions?
An important technique for determining whether or not a curve is the graph of a function
y = f (x) is called the vertical line test. For the graph of a function, we know that each
x-value corresponds to exactly one y-value, so if we draw a vertical line through a given xvalue, this line will intersect the graph precisely once. Conversely, any curve which satisfies
this vertical line test can be realized as the graph of a function y = f (x).
Equations and Graphs
We’re often interested in representing relations graphically; for relations between subsets
of the real line, we can do this by plotting in the Cartesian plane R2 points (x, y) which
satisfy the relation. Sometimes we do this in a point-by-point fashion, as we would with the
relations in examples 1 and 2 above; there are many functions whose graphs you should be
able to reproduce from memory (if not now, soon), such as those in example 4 above.
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Example.
1. Draw the graphs corresponding to the relations in examples 1 and 2 above.
2. Draw the graph of the relation in example 3 and explain why x is not a function of y.
3. Draw the graphs of y = x3 , y = x3 − 2, and y = (x − 2)3 .
4. Draw the graph of the relation x2 + y 2 = 4. Does this curve determine y as a function
of x?
√
5. Draw the graphs of y = |x|, y = x, and y = 1/x. Are these functions? If so, what
are their respective domains?
Trigonometric Functions
Given a circle of radius 1 centered at (0, 0), trace out an arclength of θ along the circle. We
call the x-coordinate of the endpoint of this path the cosine of θ, written cos(θ) and call
the y-coordinate the sine of θ, written sin(θ). In this situation, θ is known as the angle
(measured in radians).
Figure 1: Some values of the sine and cosine functions.
The ratio of sin(θ) and cos(θ) is called the tangent of θ:
tan(θ) :=
2
sin(θ)
.
cos(θ)
We also define the reciprocals of these three functions as their own functions:
sec(θ) :=
1
,
cos(θ)
csc(θ) :=
1
,
sin(θ)
cot(θ) :=
1
,
tan(θ)
called secant, cosecant, and cotangent, respectively. (There are better ways to define
these six functions, but it is assumed that you have seen them before.)
Example.
1. What are the domains and ranges of the trigonometric functions defined above?
2. Find sin(3π/4), cos(10π/3), tan(−π/6), cot(13π/6).
3. Draw the graphs of y = sin(x), y = cos(x), and y = tan(x).
Notice that since cos(θ) is the x-coordinate of our endpoint and sin(θ) is the y-value, we
can obtain a right triangle by moving from the origin to (cos(θ), 0), and then to (cos(θ), sin(θ)).
Do you remember an important identity about right triangles from high school? We have
the Pythagorean Identity, which says that if the legs of a right triangle have lengths a
and b and the hypotenuse has length c, then
a2 + b2 = c2 .
We can apply this to our triangle to obtain the following important trigonometric identity:
cos2 (θ) + sin2 (θ) = 1.
(1)
This is a very important identity, from which many other trigonometric identities can be
deduced. A variety of other trig identities can be found in the front cover of your textbook.
Example. Suppose sin(θ) =
cos(θ) and tan(θ).
1
2
and tan(θ) < 0. Use equation (1) to deduce the values of
Equations Determining Lines
Recall that given two points (x1 , y1 ) and (x2 , y2 ) in R2 , we often care about the slope of
the line passing through these two points, and that the slope measures the ratio of vertical
change to horizontal change:
y2 − y1
,
slope between x1 and x2 =
x2 − x1
provided x1 6= x2 . If x1 = x2 , we don’t define a slope between this points. Why? Notice that
if we draw a line passing through (x1 , y1 ) and (x2 , y2 ) and choose a third point (x, y), the
slope between this third point and (x1 , y1 ) will be the same as the slope between (x2 , y2 ):
y2 − y1
y − y1
=
x2 − x1
x − x1
⇒
y=
y2 − y1
(x − x1 ) + y1 .
x2 − x1
So this last line gives us an equation for the line passing through (x1 , y1 ) and (x2 , y2 ), and
in fact determines y as a function of x.
3
Example.
1. Find the equation of the line passing through (2, 4) with slope 2.
2. Find the equation of the line passing through (−1, 1) and (3, 9).
Miscellany
Definition. We call a function f (x) even if f (−x) = f (x) for all x and we call f odd if
f (−x) = −x.
Note. We get the names even and odd for these functions from polynomials. Notice that
f (x) = x2 is even, while g(x) = x3 is odd. Also notice that, unlike the integers, there exist
functions which are neither odd nor even. What can you say about a function which is both
odd and even?
Example. Are sin(θ) and cos(θ) even, odd, or neither? What about |x|, x1 ,
x2
?
x2 +1
Remember that we sometimes define functions by using different rules for different parts
of the real line; we call these piecewise-defined functions.
Example. Draw the graph of y = f (x), where

−3 − x,
x < −1
√
f (x) :=
1 − x2 , −1 ≤ x < 1 .

x − 3,
1≤x
Finally, it’s very important that you remember how to find the solutions of a quadratic
equation. Given a quadratic equation in standard form
ax2 + bx + c = 0,
there are two solutions, given by the quadratic formula
√
−b ± b2 − 4ac
x=
.
2a
(When we say that there are two solutions, we’re counting multiplicities and including the
complex solutions, which occur when b2 − 4ac < 0. There may be two, one, or zero real
solutions.)
Example.
1. Find the real solutions of x2 − 6x − 7 = 0.
2. Find the real solutions of x = 1/x + 1.
3. Find all solutions of x2 + 4 = 0.
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