Study the graph of the straight line below. Notice that for each value of x, there is exactly one value of y. This type of relation is called a function.
Example
Determine whether each relation is a function. Explain your
answer.
1. {(5, 2), (3, 5), (2, 3), (5, 1)}
Alternative Solutions :
Since
each element of the domain is paired with exactly one element of the
range, this relation is a function.
2.
Alternative Solutions :
The
mapping between x and y represents a function since there is only
one
corresponding element in the range for each element of the domain.
It
does not matter if two elements of the domain are paired with the same element
in the range.
3.
Alternative Solutions :
The graph
represents a relation that is not a function. Look at the points with the ordered
pairs (4, –3) and (4, 4). The member 4 in the domain is paired with both –3 and
4 in the range.
To determine whether an equation is a function, you can use
the vertical line test
on the graph of the equation. Consider the graph below.
To perform the test, place a pencil at the left of the graph
to represent a vertical line. Move it to the right across the graph.
For each value of x, this vertical line passes through
exactly one point on the graph. So, the equation is a function.
Example
Use the vertical line test to determine whether each relation
is a function.
4.
5.
Alternative Solutions :
The
relation in Example 4 is a function since any vertical line passes through no
more than one point of the graph of the relation. The graph in Example 4 is the
graph of the function y = |x|. The relation in Example 5 is not a
function since a vertical line can pass through more than one point. While the
entire graph is not the graph of a function, it can be separated into two
parts: the graph of the function y = Öx and the graph of the function y = – Öx.
Equations that are functions can be written in a form called functional notation. For
example, consider the equation y = 3x + 5.
equation functional
notation
y = 3x + 5 f(x) = 3x + 5
In a function, x represents the elements of the
domain, and f(x) represents the elements of the range. For
example, f(2) is the element in the range that corresponds to the member
2 in the domain. f(2) is called the functional value of f for x = 2.
You can find f(2)
as shown at the right. The ordered pair (2, 11) is a solution of the function
of f.
f(x) = 3x + 5
f(2) = 3 ⋅ 2 + 5 Replace
x with 2.
f(2) = 6 + 5
or 11
Example
If f(x) = 2x + 3, find each value.
6. f(4)
Alternative Solutions :
f(x) = 2x + 3
f(4) = 2(4)
+ 3 Replace x with 4.
= 8 3 Multiply.
= 11 Add
7. f(6a)
Alternative Solutions :
f(x) = 2x + 3
f(6a) = 2(6a) + 3 Replace
x with 6a.
= 12a + 3 Multiply.
Functions
are often used to solve real-life problems.
Example
Anatomy Link
9. Anthropologists
use the length of certain bones of a human skeleton to estimate the height of
the living person. One of these bones is the femur, which extends from the hip
to the knee. To estimate the height in centimeters of a female with a femur of length
x, the function h(x) = 61.41 + 2.32x can be used.
What was the height of a female whose femur measures 46 centimeters?
Alternative Solutions :
h(x) = 61.41 + 2.32x Original
equation
h(46) = 61.41
+ 2.32(46) Replace x with 46.
= 168.13 Simplify.
The woman was about 168 centimeters tall.
Sumber
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