A function is a rule
assigns to every element in a set D (domain) exactly one element in the set R (range). We will
functions as a set of ordered pairs (x,y) where x is in the domain and
is in the range with
y = f(x).
Finding the domain of a
function from its rule is not difficult. Look at the examples
1) Give the domain of each.
a) f(x) = 3/(x - 5) The domain consists
numbers for x that are defined for the function. Since the
function doesn't exist at x = 5 ( it makes the denominator 0), the
domain is all real
numbers but 5.
b) f(x) =
Since 4 - x is under the radical,
- x must be greater than or equal to zero, otherwise the answers will
imaginary. To find the domain, solve the inequality 4 - x >
-x > -4
x < 4. Thus, all numbers less than or equal to 4
represent the domain for this function.
When trying to find the domain and range from a graph, the domain
found by looking at the graph from left to right. The range is
by looking at the graph from top to bottom. Find the domain and range
the given functions.
Look at the graph
from left to right. For the x values, they start at -1 and end at
1. So the domain is -1 < x < 1. Look
at the graph from
top to bottom. The high point is 1 and the low point is 0.
the range is 0 < y < 1.
The graph above has
a domain that is all real numbers. It doesn't stop to the right
left. The range of the function is y > 0. The
is zero and it has no high point.
To determine if a
graph is a function or not, we can use the vertical
line test. If no vertical line
crosses a graph more than once, then the graph is a function.
Study the graphs
below and determine if they are functions.
The top two are not
functions because a vertical line passed through the graphs will cross
the circle and
parabola twice. The bottom two are functions because vertical
passed anywhere through these graphs will only cross once.
When working with
functions, the x variable is the independent variable and f(x) is the
dependent variable. The function depends on the values you pick
for the x value!!
Let's head on to
the next section!!