VOCABULARY CHAPTER 1
DOMAIN: All X-values of a function that produce a definable solution (eg 0 is NOT in the domain of f(x) =
1/x, since 1/0 produces an undefinable solution).
RANGE: All Y-values that a function takes on (Eg. The function f(x) = e^(x)-the exponential function-has
range (0, infinity), since the solutions to f(x) = e^(x) can never have an answer less than or equal to 0).
CONTINUITY:
The graph of a function is considered continuous if it is definable at every X value within its domain.
There are three types of discontinuous graphs:
-Removable Discontinuity:
The function f(x)= (x^(2) - 1)/(x – 1) has a removable discontinuity at 1, since it is undefinable at that
single point, producing what appears to be a “hole” in an otherwise continuous line. This discontinuity is
called removable because mathematicians can “force” a single point into the graph to make the graph
continuous, and essentially remove the removable discontinuity. For instance, they would define our
function f(x) as:
D: { f(x)= (x^(2) - 1)/(x – 1) for all x not equal to 1}
{ f(x) = 2 when x = 1}
Though we know this is not true, we simply accept it as fact for the sake of certain operations you will
most likely learn later in math (like calculus!)
-Infinite Discontinuity:
The function f(x) = 1/x is not only undefined at 0, but stretches off to infinity on its range as the x-values
approach zero from either side. This is because as you pick values closer to zero (such as x = 0.0001) your
solution becomes infinitely large (f(0.0001) = 10,000), and goes off to infinity (hence the name). A similar
situation happens from the left side of the graph, getting smaller negative values and larger negative
solutions. When a graph stretches off to infinity as a result of a discontinuity, it creates what we call an
asymptote, which will be defined next.
-Jump Discontinuity:
The function f(x)=int(x) is an example of the relatively less common Jump Discontinuity. Though this
graph is technically defined at all points in its domain, it is not defined at all points in its range, as the
graph “jumps” from one y-value per integer to the next, creating a noticeable vertical gap between
integer values.
The last 2 of the 3 discontinuities cannot be made continuous by adding a single point, so they are called
non-removable discontinuities.
Asymptote:
An asymptote is a technically invisible line that defines a boundary on a graph. There are two types of
asymptotes:
-Vertical Asymptote
As stated before, the graph of f(x)=1/x is undefined at x = 0, creating a graph that stretches off to infinity
on both sides at x = 0. Thus, 1/x has a vertical asymptote at x=0. If you graph this on your calculator, you
might see a vertical line
-Horizontal
Horizontal asymptotes occur on graphs that are either bounded above or below. f(x)=1/x has a
horizontal asymptote at y = 0. The logistic function f(x)= 1/(1 + e^(-x)) has a horizontal asymptote at y = 1
and y = 0. Despite being bounded above and below, however, f(x) = sinx and f(x) = cosx does not have
horizontal asymptotes. This is a rare exception. In order for the boundary to have a horizontal
asymptote, there must be a boundary as x goes to positive or negative infinity.
Increasing and Decreasing
A graph is consider increasing between two sets of coordinates (x(1), y(1)) and (x(2), y(2)) if the y(2)
coordinate is larger than the y(1) coordinate. For example, the graph of y = x^(2) is considered increasing
from (1, 1) to (3,9) and decreasing from (-2,4) to (0,0) for similar reasons.
Boundaries (Boundedness)
A boundary is a value (x OR y) on a graph that a graph cannot pass. For this class, we will mainly use this
term to describe vertical bounds, such as f(x)=e^(x), which has a vertical boundary (horizontal
asymptote) at y=0. The graph not only cannot pass this value, but almost always can never even touch it,
which is why we consider the value to be a boundary point. Graphs can be bounded above (eg f(x) = (x^(2))), below (as shown previously) and both above and below (f(x)=sinx). Graphs that are bounded
above and below are simply considered to be bounded.
End Behavior and Limit Notation
The end behavior of a graph is an analysis of what happens to a graph as the x-values increase to
infinity. We describe the y-value that the graph reaches as, “The limit of f(x) as x approaches infinity.” For
Instance, the limit of f(x) = x as x approaches infinity is positive infinity. The limit of f(x)=1/x as x
approaches infinity is 0. The limit of f(x)= 1/(1 + e^(-x)) as x approaches positive infinity is 1, while the
limit as x approaches negative infinity is 0.
Composite Functions
When we denote a function f(x) = x^2 by f(2), it means we replace each instance of x in the equation with
2, thus we get f(2) = (2)^2. If we were to define another function g(x) to be x + 2, then we can replace
each instance of x in f(x) with our new expression x + 2, getting the composite function f(g(x))=(x+2)^2.
We usually write f(g(x)) as f o g, which is read as, “F of G.” Composite functions are then essentially
functions within functions, or a function where each instance of x (or whatever variable is being used)
with another function, and the variable used in the composite becomes the new variable you are
ultimately working with (for this reason, most of the composite functions you’ll encounter will use the
same variable for both functions you use).
-COMPOSITE FUNCTIONS AND DOMAIN
When dealing with composite functions, it is crucial to take exceptional note of their domain. When
dealing with a function within a function, you are subject to restrictions within BOTH of their domains.
For instance, f o g where f(x) = x^3 and g(x)=squareroot(x-4) gives you f(g(x)) = (squareroot(x-4))^3.
While normally there are no domain restrictions on x^3, there is a restriction on squareroot(x), namely
that x cannot be less than 0, or in our case, x-4, x must be 4 or greater. Therefore the domain of this
function would be [4,positive infinity), even though the parent function, f(x) = x^3 has a domain of
(negative infinity, positive infinity). As usual though, just be vigilant about spotting zero’s in the
denominator and negative square roots, other than that, composite functions will rarely trick you.
A few Operation Rules for Functions:
(f + g)(x) = f(x) + g(x)
(f - g)(x) = f(x) - g(x)
(f g)(x) = f(x) g(x)
(f/ g)(x) = f(x) /g(x), provided g(x) does not equal 0.
The domain of these functions consists of all numbers that belong to both f and g at the same time.
Parametric Equations
Parametric equations are functions in three variables. What this means is that you have a third variable
(usually we use t) to define the movement of two variables, x and y in relation to another variable, which
you will most often see defined as t. As far as you need to know now, there are two skills you need to be
competent with for parametrics:
1. Being able to solve parametric equations for designated values of t
FOR INSTANCE:
consider the parametric equation:
x=t+1
y = 2t – 4
Solving for t = 0,1,2 and 3, we get the x-y coordinate pairs
(1,-4), (2, -2), (3, 0), and (4, 2), respectively.
2. Being able to convert a parametric equation into standard form by eliminating t from the equation.
We can do this using substitution. Let’s look at our example equation once again:
x=t+1
y = 2t – 4
If we subtract 1 from both sides of our x equation, we end up with
t=x–1
we now have an equation for t. Let’s plug our equation into our y = 2t – 4 for every instance of t:
y = 2(x-1) – 4
y = 2x – 2 – 4
y = 2x – 6
now if we plug in 1, 2, 3, and 4 for x, we get the following coordinate pairs:
(1,-4), (2, -2), (3, 0), and (4, 2), respectively.
We see now that converting the same equation to standard form yields the same answers. On problems
without specific instructions, using either form for solving the equation is acceptable. If they ask you to
convert into standard form, use method 2. If they just list off values, use method 1.
Inverse Functions
The inverse function is defined algebraically as f^(-1)(x) (“f to the power of negative one of x.”), and on a
graph, is a reflection of its original function across the line y = x. One of the most familiar examples is f(x)
= squareroot(x), which is the inverse function of x^(2). When both of these graphs are displayed with the
function y = x (you can graph these on your calculator to verify this), you will see one half of the parabola
x^(2) mirrored across the line y = x. That mirror is y = squareroot(x), the inverse of x^(2).
Finding the inverse algebraically:
To find the inverse of a function f(x), do the following steps:
1. Rewrite the equation in x – y notation.
2. Switch every instance of x with y and every instance of y with x.
3. Solve for y.
Make sure to note the domain restrictions of both your original function and your inverse function to
look for discontinuities.
Find the inverse graphically:
If you are only given a graph of a function that is not defined algebraically, you can graph the inverse by
identifying some coordinates on the original function, switching their x and y values, and graphing the
results.
Verifying that two functions are inverses of each other:
If you are given two functions and asked to find our if they are inverses of each other, simply turn them
into a composite function, f(g(x)) and g(f(x)). If BOTH equations simply to f(g(x)) = x and g(f(x)) = x, then
the two equations are inverses of each other. Remember to check both, not just one!
Piece-wise Functions
A piece-wise function is defined by multiple functions across set domains, for example:
f(x) = {2x (-infinity, 0]
x + 2 (0, 5]
1/x (5, infinity)}
Is a piece wise function, that takes on the properties of the graph 2x from negative infinity to 0, then
becomes x +2 from 0 to 5, then becomes 1/x from 5 onward.
Graphical Transformations
There are three types of transformations you need to know:
1. Horizontal and vertical translations
2. Reflections across the y and x-axis
3. Vertical and horizontal stretches and shrinks.
Horizontal and vertical translations:
If a function f(x) is instead written as f(x + n), it is translated to the left n units. For instance, f(x) = (x +
2)^(2) is the graph f(x) = x^(2) translated left 2 units.
If a function f(x) is instead written as f(x - n), it is translated to the right n units. For instance, f(x) = (x 2)^(2) is the graph f(x) = x^(2) translated right 2 units.
Reflections across the y and x-axis:
If a function f(x) is instead written as f(-x), it is reflected across the y-axis. For instance, f(x) = (-x)^(2) is
the graph f(x) = x^(2) Reflected across the y axis. Since the reflection of this graph looks the same, it
means f(x) = x^(2) is even.
If a function f(x) is instead written as –(f(x)), it is reflected across the x-axis. For instance, f(x) = -(x^(2)) is
the graph f(x) = x^(2) Reflected across the x axis.
Stretches and Shrinks:
If a function f(x) is instead written as cf(x), it is then stretched vertically by a factor of c. For instance, f(x)
=2 (x^(2)) is the graph f(x) = x^(2) stretched vertically by a factor of 2. This can also be called a horizontal
shrink.
If a function f(x) is instead written as (c/k)f(x), it is then stretched horizontally by a factor of k/c. For
instance, f(x) =1/4 (x^(2)) is the graph f(x) = x^(2) stretched horizontally by a factor of 4. This can also be
called a vertical shrink.