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Ch 2 Section 2 – Functions
Relations and Functions:
ordered pairs: (x, y) are used to represent corresponding quantities in each.
x is the independent variable, y is the dependent variable (the output, y, depends on the
defined relation or function and the input, x).
A relation is a set of ordered pairs.
A function is a relation in which, for each value of the first component of the ordered pairs, there
is exactly one value of the second component. (For every x there’s exactly one y)
In a function, no two ordered pairs can have the same first component and a different second
component.
Examples:
a)
Class members
Birth month
b)
Class members
Pets names
Domain and Range:
In a relation, the set of all values of the independent variable (x) is the domain; the set of all of
the dependent variable (y) is the range.
To find the domain and range:
If a list of ordered pairs, the list of all x’s composes the domain, the list of all y’s composes the
range.
If a graph is given, with single plots, identify the x’s, they compose the domain; and y’s, they
compose the range.
If a graph is given, with smooth curves/lines, identify the lists of x’s and y’s for the domain and
range respectively (if a smooth curve/line, this is more easily done by “boxing the curve” and
stating the interval of x’s and interval of y’s)
Example: # 12, p 209
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Agreement on Domain:
Unless specified otherwise, the domain of a relation is assumed to be all real numbers that
produce real number output when substituted for the independent variable. (This is often called
the natural domain.)
To find the domain of a rational function, find all values which make the denominator = 0 and
exclude those.
To find the domain of a square root function, find all values which make the radicand ≥ 0 and
include those.
Examples of relations with a naturally limited domain:
y=
2
,
x−3
y = x −3
Determining Functions from Graphs or Equations:
Vertical Line Test:
If each vertical line intersects a graph in at most one point, then the graph is that of a function.
i.e., if a vertical line intersects a graph in more than one point (anywhere on the graph), then that
is not a function.
Examples:
Decide whether each relation defines a function and give the domain and range.
a.
b.
Example: # 18, p 209
Identifying Functions, Domains, and Ranges from Equations:
Choose x-values in the domain, if there is exactly one y-value for the output, it is a function.
NOTE: If the relation has y to an even power, it cannot be a function (because to solve for y, you
always have two even roots, positive and negative).
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Domain: If an equation is given, determine the natural domain (any natural limitations, see
Agreement and examples above). To find the range, test various input for x. You also might try
graphing the relation to help determine the range.
Variations of the Definition of Function:
1. A function is a relation in which, for each value of the first component of the ordered
pairs, there is exactly one value of the second component.
2. A function is a set of ordered pairs in which no first component is repeated.
3. A function is a rule or correspondence that assigns exactly one range value for each
domain value. (For every x there’s exactly one y)
Examples: # 24, # 26, # 36, & # 38, p 210
Function Notation: When a function f is defined with a rule or an equation using x and y for the
independent and dependent variables, we say “y is a function f x” to emphasize that y depends on
x. We use the notation
y = f(x),
called function notation, to express this and read f(x) as “f of x” (this is NOT multiplication).
The letter f stands for function.
NOTE: f(x) is just another name for the dependent variable, y.
One advantage of this notation is that when a relation is written using this, we know it’s a
function without having to check.
Using function notation:
f(2) means find y when x = 2,
plug 2 in for every input variable, x, in the function rule or equation.
f(q) means plug q in for every input variable, x, in the function equation.
f(a + 1) means plug (a + 1) in for every input variable, x, in the function equation.
f(x + h) means plug (x + h) in for every input variable, x, in the function equation.
TAKE CARE – KEEP INPUT PARENTHESIZED.
Examples: # 48 & # 52, p 210
If a function f is defined by an equation with x and y, not with function notation, use the
following to find f(x).
Finding an Expression for f(x):
1. Solve the equation for y.
2. Replace y with f(x).
Example: # 64, p 211
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Increasing, Decreasing, and Constant Functions:
Informally:
A function increases on an interval if it rises from left to right on the interval.
A function decreases on an interval if it falls from left to right on the interval.
A function is constant on an interval if it’s graph is a horizontal line segment on the interval.
Intervals here are based on the x- values where the y-values either increase, decrease, or are
constant.
Formally:
Suppose that a function f is defined over an interval I. If x1 and x2 are in I,
a) f increases on I if, whenever x1 < x2, f(x1) < f(x2);
a) f decreases on I if, whenever x1 < x2, f(x1) > f(x2);
a) f is constant on I if, for every x1 and x2, f(x1) = f(x2);
Process:
First decide the endpoints of the interval and find their x-values to define the interval itself.
Here, that means find all relative maximums, relative minimums, flat places.
To find a relative maximum or minimum, graph the function on your grapher. Use that to locate
those values (in calculus you will use another technique based on calculus).
Then, to decide whether a function is increasing, decreasing, or constant on an interval, decide
what does y do as x goes from left to right.
Example: # 78, p 212
Example: For the following function, graph the function on your grapher. Determine the
intervals of the domain for which each function is (a) increasing, (b) decreasing, and (c) constant.
f(x) = x3 + 5x2 + 2x – 5
y
f(x) = x3 + 5x2 + 2x – 5
x
Example: # 82, p 213
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