MATH 137 Midterm Review
October 23rd 2010
Fall 2010 FUNCTIONS AND A BSOLUTE VALUE
A function f, assigns exactly one value to every element x. We can think of x as the functions
input and y as its output. For our purposes, we can use y and f(x) interchangeably. In Calculus 1,
we deal with functions taking elements of the real numbers as inputs and outputting real
numbers. We are primarily concerned with using graphs as visual representations of functions.
Here are some definitions that are useful to keep in mind:
Domain: The set of elements x that can be inputs for a function f
Range: The set of elements y that are outputs of a function f
Increasing Function: A function is increasing over an interval A if for all, the property
( ) ( ) holds.
Decreasing Function: A function is decreasing over an interval A if for all , the property
( ) ( ) holds.
Even Function: A function with the property that for all values of x: ) ( )
Odd Function: A function with the property that for all values of x: ) ( )
A function is neither even nor odd if it does not satisfy either of these properties. When
sketching, it is helpful to keep in mind that even functions are symmetric about the y-axis and
that odd functions are symmetric about the origin (0, 0). A BSOLUTEVALUE
Properties and Rules:
Example. Given that show that | |
First, we split the fraction and apply the triangle inequality to obtain:
| | | | | |
Note that for any value of x. Therefore, if we replace the denominator with 5, we
are shrinking it and thereby making the entire rational expression larger. Hence:
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After applying properties of absolute value, we can obtain the following expression:
| | | |
By applying the definition of absolute value, we get that |-1| = 1 and |5| = 5. Since |x| < 2, as
provided in the question, we can safely substitute 2 for x in the expression. Thus, we obtain:
| | | | | |
| | SKETCHING T HE USE OFC ASES
Sometimes you may be asked to sketch a function that involves piecewise definitions.
Generally, we start by looking for the key x-values where the function changes value. Then, we
use these x-values to create different cases, where we analyze the function over a particular
interval. We can apply similar methods to sketch implicitly-defined inequalities.
Example. Sketch ( ) ( ) | |
Start out by looking for key points. Applying the Heaviside definition to H(x + 1), we can see that
H(x + 1) = 1 if , or Similarly, H(x + 1) = 0 if x < -1. Next, by the definition of
absolute value, we have that the key point for |x| is at x = 0.
Use these points to establish 3 cases:
In case 1, we have ( ) ( ) .
In case 2, we have ( ) ( ) .
In case 3, we have ( ) ( )
We finish by sketching each of these lines on the appropriate x-intervals.
Example. Sketch the inequality | | .
We have two cases to consider. Our goal is to get inequalities where we isolate y.
Case 1: , which implies that
o We have , or
Case 2: , which implies that
o We have ( ) , or
To finish, sketch each of the lines defining the y-regions and shade in their intersection.