Topic 1.2_ MATHX402-032 Math for Management
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Topic 1.2: MATHX402-032 Math for Management
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Topic 1.2
Return to Module 1
(https://onlinelearning.berkeley.edu/courses/2072192/pages/module-01)
Topic 1.2 Functions
What is a function?
A function is simply a relationship. The function relates inputs to outputs, taking the input elements
from a set (the domain
) and generating outputs (together called the range
) in a set. Commonly,
elements of the domain are denoted by x
and elements of the range are denoted by y
. Not
surprisingly, just as the water pressure on your ears depends on how deep you dive, the output
elements of the range depend
on the input elements of the domain. For this reason, we call the y
variable the dependent
variable, and the x
variable the
independent
variable. This aspect of
dependency will become clear with a few examples, below.
So, in sum, there are 3 parts to a function:
The 3 Parts of a Function
The Input
The Relationship
The Output
Domain
Independent
X variable
Range
Dependent
Y variable
A good place to start in thinking about a function is to imagine two numerical or non-numerical related
things. Numerical examples might include (a) summer temperatures and lemonade sales, (b) rates of
heart disease versus obesity, or (c) SAT score compared to the weight of a high school senior’s
backpack.
Example 1.2.1
For each of these 3 examples, think of which variable depends on which, and label them independent
(
x
) versus dependent (
y
).
a. summer temperatures and lemonade sales
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b. rates of heart disease versus obesity
c. SAT score compared to the weight of a high school senior’s backpack.
Example 1.2.1.a
Lemonade sales would depend on summer temperatures, not the other way around. Therefore,
lemonade sales are the y
, or dependent
variable, and the set of y
outcomes defines the range of the
function. Summer temperatures are the x
, or independent variable, and the set of x
outcomes is the
domain of the function.
Example 1.2.1.b
You try it! Label the independent (
x
) and dependent (
y
) variable in the relationship between heart
disease and obesity.
Click to reveal the solution.
Example 1.2.1.c
You try it! Label the independent (
x
) and dependent (
y
) variable in the relationship between SAT
score and backpack weight.
Click to reveal the solution.
Non-numerical examples can also represent the logic of a function. We can define a function that has
as its domain the set of people on Earth, and its range is the set of possible heights. If you feed in a
person to the function, it will spit out the height of that person. For example, if we denote our height
function by H
, we might say that H(Kobe Bryant) = 78, or H(Elvis)
= 72. Note that functions are not
always reversible: if you try to run this machine backwards, by feeding a height in, you can't retrieve
which person that height came from, since there is more than one person of that height: since
H(Kobe Bryant)
= 78 and H(Michael Jordan)
= 78, if we feed 78 in backwards, we don't know if we
should get Kobe or Jordan. However, some functions are reversible -- they are called bijective
functions
. These functions will have an inverse function
; that is, the function that does the exact
opposite.
In order for a relationship to qualify as a function, every input can only produce one output (each x
variable can associate with only one y
variable). In the context of our height function, this simply
means that every person can only generate one height. Under this constraint, our height function
qualifies as a function.
You might have gathered from the real-world examples we have presented here that some data
(such as atmospheric water pressure versus depth) is precisely related, some (like obesity and heart
disease) is correlated, but not exactly, and some (such as our height data) doesn’t follow a pattern at
all. It is the language of statistics (modules 3-7) that helps us to find patterns and quantify these
correlations and relationships into equations. When functions follow predictable patterns, and can be
represented by equations and graphs, they are used to model real-world phenomena, in order that
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Topic 1.2: MATHX402-032 Math for Management
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we can understand our world better and make predictions more accurately. This is why functions are
widely applied in fields like physics, engineering, astronomy, statistics, economics, and finance, just
to name a few.
Graphing functions
Inputs and their matching pairs are together called ordered pairs
. Therefore, functions can be seen
as sets of ordered pairs. Naturally, most functions we deal with in mathematics use sets of numbers
as their domain and range, and the relationship part of the function is expressed as an equation.
We graph functions using the Cartesian coordinate plane
, plotting the ordered pairs on the 2 axes.
Though the axes are commonly known as the x
-axis and the y
-axis, the variable names like x
and y
may change (for example, in Economics, when we graph Supply and Demand, we label our axes
“Price” and “Quantity”). On the horizontal axis, we typically label elements of the independent/input
domain, and on the vertical axis, we label elements of the dependent/output range.
Linear functions
We will start with the simplest function, the linear function. As its name suggests, this function graphs
to the shape of a line, and its equation takes the form , also . It is
common to have stand in for since the y-axis corresponds to the values that takes on
(the range). The represents the slope
, or slant, of the line, and the represents the y-intercept
(where the line crosses the y-axis). Hence the name of this equation form, the slope-intercept form
of a linear equation.
A second form of a linear equation (same substance, just in a different form), is created using the
slope and one point on the line , hence, its name: the point-slope form
of a linear
equation. Its general form looks like .
Slope is calculated as the “rise/run” or “change in y’s/change in x’s”, and can be derived from any
2 points on a line. Using the points and to present a general equation for slope:
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Example 1.2.2
Find the equation of the line passing through (2, 1) and (3, 4).
Our first step is to calculate the slope:
and then plug in one of the points to the point-slope form of the equation for a line:
Solving for y
gives us , so the function form of our line is , a line with a
slope of 3 and a y-intercept of -5. We will graph this equation as part of the next example.
For any function, to understand the relationship between an equation and its graph, it is easiest to
start with the simplest form, what is sometimes called the “parent function.”
Example 1.2.3
The very simplest form of a linear function is . Expanded to form, this equation
becomes . We can graph this function by plotting points: (-1, -1), (0, 0), (2, 2), etc., or
by using the y-intercept of zero and slope of 1 to draw the graph.
Linear “parent” function
Red
: Now, let’s consider our function from the previous example . The slope of has the effect of increasing the slant of the line from the red
to the blue
line. The y-
intercept has the effect of shifting the line down from the blue to the green
.
Shifts to Linear Functions
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Red
: Blue
: Green : Example 1.2.4
Find where the lines and intersect.
Ask yourself: First, let’s think about this question and ask ourselves: How do we know these lines
actually cross? (Hint: What would have to be true if they did not cross?) Click to reveal.
If these lines did not cross, they would have to be parallel lines. If that were the case, their slopes
would be equal. With unequal slopes of 3 and -9, we know that, somewhere, these lines will intersect
each other.
What is true at the point of intersection of these 2 lines is that the same ordered pair (x, y) will satisfy
both
equations. To find that point, we can simply set the 2 equations equal to one another (because
the y
’s are the same) and solve for a single x
(because we are finding the common x
):
Therefore, the lines cross when
. The function value at this point is
, so the lines cross at
.
Intersection
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Red
: Blue
: Quadratic functions
Quadratic functions (2nd degree polynomial functions in the form: ) are
frequently useful in modeling real-life optimization scenarios in business, economics, and finance. As
with linear functions, we will review and highlight the features of quadratics by starting with the parent
function, a quadratic in its simplest form: , or simply, .
Graphing the parent function is a simple matter by plotting points, resulting in an upward facing
parabola with its vertex at the origin (0,0). Note that the vertex is the minimum
of the upward facing
parabola.
Red
:
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Now, let’s review ways that a quadratic function will shift and change from the parent function. It is
useful to understand how changes in the quadratic equation translate graphically since this may help
us to interpret an application problem. Below, observe the change or shift in the graph of the
parabola, compared to the difference you see in each equation, from the parent equation ,
always graphed in red
:
Change to the “a” coefficient Red
: Blue
: Green
: Change to the “x
” term
Red
: Blue
: Green
: Adding a constant term
Negative “a” coefficient
Red
: 2
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Red
: Blue
: Green
: Blue
: Solving Quadratic Equations
Sometimes, we will need to solve quadratic equations. We may be solving for:
Where the parabola crosses the x-axis (x-intercepts, or “zeros” of the function), in which case we would set the
function equal to zero, since this is the place where , the y-value, equals zero;
An intersection of a line or other function with the parabola, in which case we would set the
functions equal to one another (like we did in Example 1.2.4 for linear functions), since they have
the same values at that point; or
A maximum or minimum, in which case we will be finding the coordinates of the Vertex of the
downward (max) or upward (min) facing parabola.
To solve a quadratic equation for zeros, because it is second degree, you will expect (in most cases)
2 solutions. This makes sense, considering that the parabola (in most cases) will cross the x-axis in 2
places. When you took Intermediate Algebra, you learned that the process for solving a quadratic
equation followed these steps:
1. Attempt to factor the quadratic by first factoring out any common terms to simplify, then factoring the equation
(“factors of c that add to b”) to find solution(s);
2. If the equation cannot be (or is difficult to) factor, you can find solutions using the quadratic
formula, which uses the a, b and c coefficients from the form of the
function.
The Quadratic Formula
Example 1.2.5
Find the x-intercept(s) of the function .
To find the x-intercepts of this function, we will set it equal to zero, since the x-intercepts are the place
where , the y-value, equals zero:
This quadratic equation is easily factorable into a perfect square:
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In order for this equation to be true:
Then must be true, because you can’t get a result of zero unless zero is one of the
multipliers, so:
...and there is only one solution. This means that the graph must sit on the axis on its vertex.
The graph that results is labeled and illustrated in blue in the above set of graphs
entitled: Change to the “ ” term .
Example 1.2.6
Find the x-intercept(s) of the function .
Due to the “a” coefficient (of 2) and the lack of easy factorability, we will employ the quadratic formula
in this case.
, , and the x-intercepts are:
and Graphically, this means that the parabola crosses the x-axis in 2 places.
The Vertex Formula -- a “Shortcut”
A very handy shortcut to finding the vertex of a parabola (and therefore solving for a maximum or
minimum value in an application problem) is illustrated through the examples we just completed. The
secret is in the discriminant, which is the part of the quadratic formula,
, that is contained
under the radical sign.
In Example 1.2.6, the discriminant is , giving rise to 2 solutions
through use of the quadratic formula.
In Example 1.2.5, we had only one solution to our equation. Had we used the quadratic formula, we
would have found that the discriminant is . If you look at the quadratic
formula, you can see why this is so. A discriminant of zero essentially eliminates the term, leaving a single solution equal to
, because
. Note that this is
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also the location of the vertex of the parabola, and that this will always be so, since this is the only
location on the parabola where a single solution is found.
Therefore, we have found a new shortcut for locating the coordinates of the Vertex (and so, the
maximum or minimum) of a quadratic function and its parabola:
Example 1.2.7
Find the Vertices of the parabolas described by the functions in Examples 1.2.5 and 1.2.6 using the
Vertex Formula “shortcut.”
From the function from Example 1.2.5 we have:
To find the y coordinate of the vertex, we plug the x value into the original equation:
So the Vertex is From the function from Example 1.2.6 we have:
You try it! Try to find the vertex of the preceding function. Click to reveal.
To find the y coordinate of the vertex, we plug the x value into the original equation:
So the Vertex is It is useful at this point to practice graphing your results. Try graphing these functions on a graphing
calculator or use an online calculator such as the one available at www.desmos.com
(http://www.desmos.com) .
Other functions
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Other functions that deserve mention and illustration simply to complete our discussion of functions
include:
1. Exponential and Logarithmic functions
. The parent form of the exponential function is , which graphs to the red
shape below. The parent form of the logarithmic function, the
inverse of the exponential function, is , which graphs to the blue shape below.
Exponential and logarithmic functions are used frequently in finance applications and warrant their
own sections. We will cover them in depth in Topic 1.3.
Parent function: Exponential and Logarithmic function
Red
: Blue
: 2. Rational functions.
These are functions in “fraction” or “ratio” form, with the variable in the
denominator. The parent function for this form of function is , which graphs to this shape:
Parent function: Rational function
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Red : Ask yourself: Why is it that this function never “reaches” ? Why is it that it never “reaches”
? Click to reveal.
In the function , by definition, or the result is undefined. This means that the y-axis
will become an asymptote. Similarly, there is no value of x whereby or y can equal zero.
Therefore, the x-axis is also an asymptote to this function.
Summary
In this section, we reviewed the concept of a function and identified several types of functions that are
relevant to this course. We related equations to their graphs and showed how changes in the
equations translate to shifts in the graphs and changes in their shape. The different types of functions
are distinguished by differences between their equations, specifically, where the variable resides. A
variable in the exponent, for example, identifies an exponential function, and a second-degree
polynomial identifies a quadratic function. Being comfortable with the language of functions gives a
conceptual grounding to the topics we cover in this course.
Return to Module 1
(https://onlinelearning.berkeley.edu/courses/2072192/pages/module-01)
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