Download function - croninmath

Math in Our World
Section 7.6
Functions
Learning Objectives





Identify functions.
Write functions in function notation.
Evaluate functions.
Find the domain and range of functions.
Determine if a graph represents a function.
Relations
A relation is a rule matching up two sets of
objects. Relations are often represented by
sets of ordered pairs.
The following are examples of relations:
A = {(2, 3), (– 2, 1), (12, – 3), (2, – 1)}
B = {(9, 0), (– 3, 1), (3, 9), (– 1, 5), (2, 3)}
C = {(x, y) | 3x + 5y = 7}
Functions
A function is a relation in which each x coordinate
gets paired with exactly one y coordinate. In other
words, the first coordinate is never repeated with a
different second coordinate.
A = {(2, 3), (– 2, 1), (12, – 3), (2, – 1)} is not a
function because the points (2, 3) and (2, – 1) have
the same first coordinate.
B = {(9, 0), (– 3, 1), (3, 9), (– 1, 5), (2, 3)} is a
function because no first coordinate is repeated.
Functions
A function is a relation in which each x coordinate
gets paired with exactly one y coordinate. In other
words, the first coordinate is never repeated with a
different second coordinate.
C = {(x, y) | 3x + 5y = 7} is a function because for
each value of x that you substitute into the
equation, there is only one possible value of y that
corresponds to it.
EXAMPLE 1
Identifying Functions Defined
by Equations
Which of the following equations represent
functions?
(Assume that x represents the first coordinate.)
(a) y = x2 (b) 3x2 + y – 2x = 5
(c) x = y
SOLUTION
(a) This is a function. Every number has only one square, so
every value of x has only one associated y.
(b) This is also a function. Every x will again have only one
associated y.
(c) This is not a function. Positive values of x will correspond
to two possible values of y. (i.e. if x = 2, y can be 2 or – 2.)
Function Notation
The equation y = x2 represents a function
that relates variables x and y. We call x the
independent variable and y the dependent
variable because its value depends on the
choice of x.
Another way to write the same function is
f(x) = x2.
This is known as function notation, and is
read aloud as “f of x equals x squared.”
Function Notation
To write an equation in function notation,
solve for y in terms of x, then change the
letter y to the symbol f(x).
Functions can also be called by names other
than f. Any letter other than x or y is
acceptable.
EXAMPLE 2
Writing a Function in
Function Notation
Write 3x – 2y = 6 in function notation.
SOLUTION
We need to solve the equation for y, then replace y with f(x).
Evaluating a Function
When a function is written as f(x), f(2)
means to find the value of the function when
x = 2. This is known as evaluating a
function. We call x = 2 the input, and the
resulting value of the function the output.
EXAMPLE 3
Evaluating a Function
Let f(x) = x2 + 3x – 5. Find f(3), f(2), and f(0).
SOLUTION
Domain and Range
The domain of a function is the set of all
values of the independent variable x that
result in real number values for y.
The range of a function is the set of all
possible y values.
EXAMPLE 4
Finding the Domain and Range
of a Function
Find the domain and range of each function:
EXAMPLE 4
Finding the Domain and Range
of a Function
SOLUTION
There are no restrictions on what values x can be; therefore
the domain is all real numbers. Since x2 is never negative,
the range is {yy ≥ 0}.
Since the square root of a negative number is undefined, x
cannot be negative. Therefore, the domain is {xx ≥ 0}.
Since the square root of x is never negative the range is
{yy ≥ 0}.
EXAMPLE 4
Finding the Domain and Range
of a Function
SOLUTION
Since the denominator of a fraction cannot be zero, we must
exclude x = – 1. Every other x value will result in a real
number output, so the domain is all real numbers except – 1,
which we write as {xx ≠ – 1}.
The range is not obvious, but notice that an output of 3
would make the equation
Multiplying both sides by x + 1, we get the contradiction
3x + 3 = 3x – 2, so the range is {yy ≠ 3}.
Vertical Line Test
We know that a relation is
not a function if any x value
corresponds to more than
one output.
Consider the graph of an
equation shown. The two
points labeled have the
same x coordinate, so the
equation is not a function.
This indicates that if any vertical line crosses a graph
more than once, the graph does not represent a function.
Vertical Line Test
In this graph, there is no
vertical line that crosses
the graph more than once,
so it is the graph of a
function.
Vertical Line Test for Functions
If no vertical line can intersect
the graph of a relation at more
than one point, then the
relation is a function.
EXAMPLE 5
Using the Vertical Line Test
Use the vertical line test to determine whether each
relation graphed represents a function.
EXAMPLE 5
Using the Vertical Line Test
SOLUTION
This is a function since any
vertical line will cross the
graph only once.
EXAMPLE 5
Using the Vertical Line Test
SOLUTION
This is not a function since
there is a vertical line that
crosses the graph in more
than one place.