Math 125 Section 3.1 – Functions
Def: A function consists of a set of inputs called the domain, a set of outcomes called the range and a rule by which each
input determines exactly one outcome.
This does not have to involve numbers
• Computer Keyboard – monitor
• Dating
In order for a set of numbers (or a curve) to be a function, it must pass the vertical line test. Meaning that once plotted, if
we draw any vertical line over the graph and it crosses the curve at more than one point, it is NOT a function.
EX:
•
•
•
Parabola IS a function y= x 2
Circle is NOT a function x 2 + y 2=4
(1,3), (1,-3), (2,5), (2, -5), (3, 8), (3 -8) – NOT a function
If something is a function, then there will be a specific set of allowed inputs (refer to dating as a function). With numbers,
our domain will always be all real numbers unless one of these exceptions exists:
• IF the function has a denominator, this denominator cannot equal zero
• IF the function is an even root, the radicand (the expression under the radical) must be greater than or equal to
zero
• IF the function is defined with a limited domain, we honor that limitation.
EX:
•
•
•
•
•
4
y= x
y= √ x−6
y= √ 4−x
1
y=
x+3
2
x +2
y= 2
x −3x+2
** We call x the independent variable because we can choose any value for x that in the domain. We call y the
dependent variable because it's outcome depends on the input value for x.
Therefore, we often will write f(x) for y since y is dependent on x. ((There is nothing special about “f”, it can be any
letter)).
So our above examples could be written
f ( x)= x
f (x )= √ x−6
4
f ( x)=
1
and so on
x+3
To find the values for f(x) at a given x-value, we
1. replace the “x” terms with parenthesis
2. fill these parenthesis with what they want
3. use order of operations to simplify
EX:
2
g ( x)=−x +4x – 5
find g(1), g(-1), g(a), g(x+h)
DO p. 140 (2, 4, 12, 16, 20, 23, 25, 34)
Math 125 – Section 3.2 – Graphs of Functions
This section will feature graphing special functions
1.
2.
3.
4.
5.
Lines
Piecewise Linear Functions
Absolute Value Functions
Step Functions
Square Roots
1. Lines
To graph a line, we need two points. The easiest to get is the y-intercept (when x = 0) and then one other point.
ex.
f ( x)=3x−2
X | Y
0 | -2
1 | 1
Plot these two points and connect them!
2. Piecewise Linear Functions
{
f ( x)= x−2 if x≤3
−x+8 if x>3
Here, we simply need to draw two lines. The first point we pick is the one given in the pieces (x = 3)
Then we pick one point from the left and right of that value.
Line One (
x |
3 |
0 |
x≤3 ):
y
1
-2
Line Two (
x |
3 |
5 |
x>3 ):
y
5
3
Now graph these within their domains.
3. Absolute Value Functions
An absolute value problem is a special case of the piecewise function.
ex.
f ( x)=∣x−1∣
Here the break point is when
graph from there.
x−1=0 or x=1 So, we try three points, x = 1, x = 0 and x = 2 and plot the
4. Step Function
The function f ( x)=[ x ] is a special function. I will look like a stair step. The brackets here mean “the
greatest integer LESS than x. We fill in a chart until we have the basic pattern:
x | [x]
0
0
½
0
¾
0
1
1
3/2 1
etc.
5. Square root functions
These are half of a “horizontal parabola”. We first get the domain, and then need only ONE more point in the
domain.
Example: g ( x)= √ x−1
Here, the domain is that
x | g(x)
1
√ 1−1=0
5
√ 5−1=2
Plot these points.
DO p 151 (2, 3, 10, 18, 28)
x≥1