GEN.

MATH FIRST QUARTER REVIEWER  (2;8]

KEY CONCEPTS OF FUNCTIONS It is obvious that 2 is the lower number and 8 the upper number. The
round bracket means 'excluding 2', since xx is greater than 2, and the
Relation- a rule that relates value from a set of values (domain) to a set square bracket means 'including 8' as x is less than or equal to 8.
of second values (range)
PIECEWISE FUNCTION
Function- a relation where each element in the domain is related to
only one value in the range by some rules Functions assign outputs to inputs. Some functions have simple rules,
-a set of ordered pairs(x,y) such that no two ordered pairs have the like "for every x, return x²." However, there can be other rules that are
same x-value but different y-value more elaborate. For example, "If x<0, return 2x, and if x≥0, return 3x."
These are called *piecewise functions*, because their rules aren't
Dependent and Independent Variables uniform, but consist of multiple pieces.

Thus far, all the graphs you have drawn have needed two values, an x- ****REVIEW: LINEAR EQUATIONS****
value and a y-value. The y-value is usually determined from some
relation based on a given or chosen x-value. These values are given How to graph linear equations
special names in mathematics. The given or chosen x-value is known as
the independent variable, because its value can be chosen freely. The Step 1: Put the equation in Slope Intercept Form y=mx+b
calculated y-value is known as the dependent variable, because its
value depends on the chosen x-value.
Step 2: Graph the y-intercept point (the number in the b position) on
the y-axis.
Domain and Range
Step 3: From the point plotted on the y-axis, use the slope to find your
The domain of a relation is the set of all the x-values for which there second point. Remember, the slope is the number in the m position in
exists at least one y- value according to that relation. The range is the your equation.
set of all the y- values, which can be obtained using at least one x-value.
m=slope (rise/run)
If the relation is of height to people, then the domain is all living people,
y 2− y 1
while the range would be about 0,1 to 3 metres — no living person can m(given the coordinates)=
have a height of 0m, and while strictly it's not impossible to be taller x 2−x 1
than 3 metres, no one alive is. An important aspect of this range is that
it does not contain all the numbers between 0,1 and 3, but at most six Step 4: Draw your line using the two points you plotted (y-intercept
billion of them (as many as there are people). (b) first, slope (m) second. Be sure your line is pointing the right way.

These are two ways of writing the domain and range of a function, set
notation and interval notation.

Set Notation

A set of certain x-values has the following form:

x:conditions, more conditions

We read this notation as “the set of all x values where all the conditions
are satisfied”. For example, the set of all positive real numbers can be
written as {x:x∈R,x>0} which reads as “the set of all x values where x is a
real number and is greater than zero”.

Interval Notation

Here we write an interval in the form 'lower bracket, lower number, Writing Linear Equations Given a Graph
comma, upper number, upper bracket'. We can use two types of
brackets, square ones [;]or round ones (;). A square bracket means Step 1: Identify the slope, m. This can be done by calculating the slope
including the number at the end of the interval whereas a round between two known points of the line using the slope formula.
bracket means excluding the number at the end of the interval. It is
important to note that this notation can only be used for all real
Step 2: Find the y-intercept. This can be done by substituting the slope
numbers in an interval. It cannot be used to describe integers in an
and the coordinates of a point (x, y) on the line in the slope-intercept
interval or rational numbers in an interval.
formula and then solve for b.
So if x is a real number greater than 2 and less than or equal to 8, then x
is any number in the interval
Step 3: Once you've got both m and b you can just put them in the
equation at their respective position.
****END OF LINEAR EQUATIONS REVIEW****  h(x) = (2x + 1) / (x + 3)

The Absolute Value Function The rational functions to explored in this tutorial are of the form 

The Absolute Value Function is a famous Piecewise Function. f(x) = (ax + b)/(cx + d)
It has two pieces:

below zero: -x where a, b, c and d are parameters that may be changed, using
from 0 onwards: x sliders, to understand their effects on the properties of the graphs
of rational functions defined above.

Here are the general definitions of the two asymptotes.

 
1. The line   is a vertical asymptote if the graph increases
or decreases without bound on one or both sides of the line
as x moves in closer and closer to  .

2. The line   is a horizontal asymptote if the graph

f(x) = |x|
approaches   as x increases or decreases without bound. 

Note that it doesn’t have to approach   as x BOTH

increases and decreases.  It only needs to approach it on one
side in order for it to be a horizontal asymptote.

 
Example: piecewise function:
where n is the largest exponent in the numerator and m is the largest
exponent in the denominator.
 
We then have the following facts about asymptotes.
 
1. The graph will have a vertical asymptote at   if the
denominator is zero at   and the numerator isn’t zero
at  .

which looks like:  

2. If   then the x-axis is the horizontal asymptote.

3. If   then the line   is the horizontal asymptote.

4. If   there will be no horizontal asymptotes.
 

Process for Graphing a Rational Function

 
What is h(-1)?   x is ≤ 1, so we use h(x) = 2, so h(-1) = 2 1. Find the intercepts, if there are any.  Remember that the y-

intercept is given by   and we find the x-

What is h(1)?   x is ≤ 1, so we use h(x) = 2, so h(1) = 2 intercepts by setting the numerator equal to zero and
solving.
What is h(4)?   x is > 1, so we use h(x) = x, so h(4) = 4
2. Find the vertical asymptotes by setting the denominator
equal to zero and solving.
RATIONAL FUNCTION
3. Find the horizontal asymptote, if it exits, using the fact
above.
A rational function is defined as the quotient of two polynomial
functions. 4. The vertical asymptotes will divide the number line into
regions.  In each region graph at least one point in each
f(x) = P(x) / Q(x) region.  This point will tell us whether the graph will be
above or below the horizontal asymptote and if we need to
Here are some examples of rational functions: we should get several points to determine the general shape
of the graph.
 g(x) = (x2 + 1) / (x - 1) 5. Sketch the graph.
Example 1  Sketch the graph of the following function.

                                                           

Solution
So, we’ll start off with the intercepts.  The y-intercept is,
                               

The x-intercepts will be,

                                 

 
Now, we need to determine the asymptotes.  Let’s first find the vertical INVERSE FUNCTION
asymptotes.
                                       The inverse of a function has all the same points as the original
function, except that the x's and y's have been reversed. This is what
So, we’ve got one vertical asymptote.  This means that there are now they were trying to explain with their sets of points. For instance,
supposing your function is made up of these points: { (1, 0), (–3, 5), (0,
two regions of x’s.  They are   and  .
4) }. Then the inverse is given by this set of point: { (0, 1), (5, –3), (4, 0) }.
 
Now, the largest exponent in the numerator and denominator is 1 and so
by the fact there will be a horizontal asymptote at the line. EXPONENTIAL FUNCTION

If b is any number such that   and   then an exponential

  function is a function in the form,
Now, we just need points in each region of x’s.  Since the y-intercept
and x-intercept are already in the left region we won’t need to get any   where b is called the base and x can be any real number.
points there.  That means that we’ll just need to get a point in the right
region.  It doesn’t really matter what value of x we pick here we just We avoid one and zero because in this case the function would be,
need to keep it fairly small so it will fit onto our graph.  
  and these are constant functions and won’t have many of the same
                          properties that general exponential functions have.
 
Next, we avoid negative numbers so that we don’t get any complex
values out of the function evaluation.  For instance if we
allowed   the function would be,

and as you can see there are some function evaluations that will give
 Okay, putting all this together gives the following graph. complex numbers.  We only want real numbers to arise from function
evaluation and so to make sure of this we require that b not be a
negative number.

Properties of 

1. The graph of   will always contain the point  . 

Or put another way,   regardless of the value

of b.

2. For every possible b  .  Note that this implies

that  .
 3. If   then the graph of   will decrease as we 2. .  This follows from the fact that  .

3. .  This can be generalized out

move from left to right.  Check out the graph of   
above for verification of this property. to  .

4. .  This can be generalized out

4. If   then the graph of   will increase as we move
to  .
from left to right.  Check out the graph of   above for
verification of this property.

5. If   then 

LOGARITHMIC FUNCTIONS

If b is any number such that 

 and   and   then,
                      

We usually read this as “log base b of x”.

In this definition   is called the logarithm form and   

is called the exponential form. 

 The requirement that   is really a result of the fact that

we are also requiring   .  If you think about it, it will
make sense.  We are raising a positive number to an exponent
and so there is no way that the result can possibly be
anything other than another positive number.  It is very
important to remember that we can’t take the logarithm of
zero or a negative number.
 
 The “log” part of the function is simply three letters that are
used to denote the fact that we are dealing with a logarithm. 
They are not variables and they aren’t signifying
multiplication.  They are just there to tell us we are dealing
with a logarithm.
 
 Next, the b that is subscripted on the “log” part is there to tell
us what the base is as this is an important piece of
information.  Also, despite what it might look like there is no
exponentiation in the logarithm form above.  It might look
like we’ve got   in that form, but it isn’t.  It just looks like
that might be what’s happening.
 
It is important to keep the notation with logarithms straight, if you
don’t you will find it very difficult to understand them and to work with
them.

Properties of Logarithms

1. .  This follows from the fact that   .