A set of ordered pairs is called a relation.
ex.
(1,2), (3,5), (7,10)...are relations.
The first components in the ordered pairs (x-coordinate) is called the domain, and the second components (y-coordinate or f(x) coordinate) is called the range.
A function is a relation were each member of the domain is reserved one and only one member of the range.
Functions are named by using single letters.
ex.
f = {(x,y) | y = x + 4 }
or
f (x) = x + 4
The equations above are representations of functions. The second example is the most commonly used representation; it is called the function notation.
The ordered pair for functions is (x, f (x)).
consider the example;
f (x) = x + 4 at x = 2.
f (2) = 2 + 4
f (2) = 6
the ordered pair is (2,6)
ex.
find the range; given the domain {2,4,6} of f(x) = 3x - 1
f(2) = 3(2) - 1 = 6 -1 = 5
f(4) = 3(4) - 1 = 12 - 1 = 11
f(6) = 3(6) - 1 = 18 - 1 = 17
the range is {5,11,17} and the ordered pairs are (2,5); (4,11); (6,17)
Find the domain and range of:
f(x) = 2 / (x-3)
Since x cannot be zero; the domain is D = { x| x ¹ 3} and f(x) can have any value; thus the range is R = { f(x) | any real number}
Graph of Functions: [ to graph functions, just change y into f(x) from graphing equations]
Graphing functions is the same as graphing equations with the exception of naming the domain and the range.
Linear Functions:
ex.
f(x) = 3x + 2:
the function is in the slope-intercept form (y = mx + b)
m(slope) = 3
y-intercept = 2
Quadratic Functions:[ f(x) = ax²+ bx + c ]
The same as graphing quadratic equations:
ex.
graph:
f(x) = 2x²+ 8x + 9
y = 2x²+ 8x + 9 replace f(x) with y
y - 9 = 2x+ 8x complete the square
y - 9 + 4 = 2( x²+ 4x + 4)
y - 1 = 2(x + 2)² parabola with vertex at (-2,1)
y = 2(x + 2)²+ 1 solve for y and replace y with f(x)
f(x) = 2(x + 2)²+ 1
Other functions:
Rules for graphing other functions:
1. Determine the domain of the graph.
2. Check for symmetry.
if f(-x) = f(x), then symmetric about the y-axis.
if f(-x) = -f(x), then symmetric about the origin.
( functions rule out the possibility that the graph has an x-axis
symmetry)
3. Find the x-intercept and y-intercept of the graph
Evaluate f(0) to find the y-intercept.
To find the x-intercept, find the value or values that will make f(x) = 0.
4. Plot some points for the graph and reflect, according to the symmetry
test.
ex.
graph:
f(x) = | x |
the domain is D = {x | x is any real number}
f(-x) = | -x | = | x | = f(x) symmetric about the y-axis.
f(0) = | 0 | = 0
f(x) = 0 at x = 0
the intercepts are at the origin.
x f(x) 1 1 -2 2 -1 1 2 2
Vertical Line Test:
Given a graph, it can be determined if it is a function or a relation.
In order for the graph to be a function, the vertical line must only intersect the graph at one and only one point.
Horizontal Line Test:
The horizontal line test implies a one to one function, meaning that there is only one value of x associated with each value of f(x).
