Download Chapter 2: Functions and Their Graphs

Chapter 2:
Functions and Their Graphs
Section 2.1
Basics of Functions and their Graphs
Definition of Relations: A relation is any set of ordered pairs. The
set of all first components is called the domain of the relation and the
set of all second components is called the range of the relation.
Find the domain and range of the following relation:
{(0, 9.1), (10, 6.7), (20, 10.7), (30, 13.2), (36, 17.4)}
Domain
Range
A relation in which each element of the domain corresponds to
exactly one member of the range is a function. A function is a
relation in which no two ordered pairs have the same first
component and different second components.
Definition of a function: A function is a correspondence from a
first set, called the domain, to a second set, called the range, such
that each element in the domain corresponds to exactly one element
in the range.
Determine whether each relation is a function:
a. {(1, 6), (2, 6), (3, 8), (4, 9)}
b. {(6, 1), (6, 2), (8, 3), (9, 4)}
Functions as Equations:
y = 3x2 – 21x + 8
We see that each x input is assigned exactly one y output. x is
called the independent variable and y is called the dependent
variable because its assigned value depends on x.
Determine whether the following equations represent functions:
a. x2 + y = 4
b. x2 + y2 = 4
Functional Notation
y = f (x)
Evaluating a function:
Evaluate: f (2)
f (x + 3)
f (-x)
Let f (x) = x2 + 3x + 5
Graphs of Functions:
x
Graph f (x) = 2x
and g (x) = 2x + 4
f (x) g (x)
-2
-1
0
1
2
The Vertical Line Test ; If any vertical line intersects a graph in
more than one point, the graph does not define y as a function of x.
Analyzing the graph of a function:
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Identifying Domain and range from a Function’s graph:
Identifying Intercepts from a Function’s graph:
Section2.2
More on Functions and Their Graphs
Increasing, Decreasing and Constant Functions
Intervals of Increase, Decrease, or Constancy
Relative Maxima and Minima
Even and Odd Functions and Symmetry
A function is even if f (-x) = f (x)
for all x in the domain of f
(The right side of the equation does not change if x is replaced with –x)
A function is odd if f (-x) = -f (x)
for all x in the domain of f
(Every term on the right side of the equation changes its sign if x is
replaced with –x)
Identify the following functions as Even, Odd, or neither
a. f (x) = x3 – 6x b. g (x) = x4 – 2x2
c. h (x) = x2 + 2x + 1
Even Functions and y-axis symmetry:
The graph of an Even
Function is symmetric with respect to the y-axis. For every (x, y) on
the graph, (-x, y) is also on the graph.
y = x2 + 6
Odd Functions and Origin symmetry:
The graph of an Odd
Function is symmetric with respect to the origin. For every (x, y) on
the graph, (-x, -y) is also on the graph.
y = x3 – x
Piecewise Defined Functions
Sketch:
f (x) =
2x + 3, x < 1
-x + 4, x > 1
Section 2.3
Linear Functions and Slope
The Slope of a Line
Def: The slope of a line passing through two distinct points
The slope of the nonvertical line through (x1, y1) and (x2, y2) is:
m=
y2 – y1
x2 – x1
x1 – x2 ≠ 0
Find the slope of the line passing through the following points:
(-3, -1); (-2, 4)
(-3, 4); (2, -2)
Possibilities for a Line’s Slope
The Point-Slope Form of the Equation of a Line; with slope m
that passes through the point (x1, y1) is:
The Slope-Intercept Form of the Equation of a Line; with slope m
and y-intercept b is:
Write the equation of a line that passes through point (-1, 3) with
slope 4. Use Point-Slope method and Slope-Intercept method.
Graph the following:
:
3
y  x2
2
The General Form of the Equation of a Line:
Ax + By + C = 0
A, B, C are reals with A, B not both zero
Write 3x + 2y – 4 = 0 in slope-intercept form
Using Intercepts to graph Ax + By + C = 0
Graph 4x – 3y – 6 = 0
Find x-intercept
Find y-intercept
Plot and Connect Points
Section 2.4
More on Slope: Parallel and Perpendicular Lines
Slope and Parallel lines
1. If two nonvertical lines are parallel, then they have the same slope
2. If two distinct nonvertical lines have the same slope, then they are parallel
3. Two distinct vertical lines, both with undefined slopes are parallel
Write the equation of a line passing through (-3, 1) and parallel
to the line y = 2x + 1
Slope and Perpendicular Lines
1. If two nonvertical lines are perpendicular, then the product of their
slopes is -1
2. If the product of the slopes of two lines is -1, then the lines are
perpendicular
3. A horizontal line having a zero slope is perpendicular to a vertical line
having an undefined slope
Write the equation of a line passing through (3, -5) and
perpendicular to the line whose equation is: x + 4y – 8 = 0
Slope is defined as a Rate of Change:
m
y rise

x run
The Average Rate of Change of a Function:
If the graph of a function is not a straight line, the average rate
of change between any two points is the slope of the secant
containing the two points.
Let (x1, f (x1)) and (x2, f (x2)) be distinct points on graph of f (x)
The average rate of change of f (x)
from x1 to x2 is:
f (x )  f (x )
x x
2
1
2
1
Find the average rate of change of
Section 2.5
Transformations of Functions
Graphs of Common Functions:
Constant
Squaring (Quadratic)
Identity
Square Root
Absolute Value
Cubic
Cube Root
Vertical and Horizontal Shifts:(Let y = f (x) and c > 0)
h(x) = f (x) + c
h(x) = f (x + c)
Combinations of Vertical and Horizontal Shifts
Reflections in the Coordinate Axes
h(x) = -f (x)
(f (x) is reflected about the x-axis)
h(x) = f (-x)
(f (x) is reflected about the y-axis)
Nonrigid Transformations: Stretching and Shrinking
h(x) = c· f (x)
If |c| > 1
f (x) is shrunk vertically
If 0 < |c| < 1
f (x) is stretched vertically
h(x) = f (cx)
If |c| > 1
f (x) is shrunk horizontally
If 0 < |c| < 1
f (x) is stretched horizontally
Graphing Using a Sequence of Transformations
Section 2.6
Combinations of Functions; Composite Functions
The Domain of a Function:
The set of real numbers for which
the function is defined. That is, all x values such that f (x) evaluates to a real
number. Exclude any numbers that make a denominator zero or that result in
the square root of a negative number.
Find the Domain of the Following Functions:
The Algebra of Functions (Arithmetic Combinations)
Sum or Difference:
(f ± g)(x) = f (x) ± g(x)
Product:
(f g)(x) = f (x) · g(x)
Quotient:
f (x) = f (x)
g(x)
g
, such that g(x) ≠ 0
Domains of Combination Functions are those real numbers
common to both functions
Find the sum of:
Composite Functions:
Composition of Functions
The composition of function f with function g is:
(f o g)(x) = f (g(x))
The domain of (f o g)(x) is the set of all x in the domain of g
such that g(x) is in the domain of f.
Decomposing functions:
Section 2.7: Inverse Functions
Recall a function is a set of ordered pairs where no two of the
ordered pairs have the same first component and different second
components.
A one-to-one (1-1) function is a set of ordered pairs where no
two of which have the same second component.
x1 ≠ x2  f (x1) ≠ f (x2)
Definition of the Inverse of a Function
Let f and g be two functions such that
f (g(x)) = x
for every x in the domain of g
g(f (x)) = x
for every x in the domain of f
Under these conditions, the function g is the inverse of the function f.
The function g is denoted by f –1. Thus,
(f o f -1)(x) = x = (f -1 o f )(x)
f (f -1(x)) = x = f -1(f (x))
The domain of f must equal the range of f –1, and the range of f must
equal the domain of f –1.
Horizontal Line Test:
A function f is a 1-1 function and has an inverse function if
and only if no horizontal line intersects the graph of f at more than
one point.
Informally Finding the Inverse of a 1-1 Function:
UNDO what the function DOES
Formally finding the Inverse of a Function
1.
Determine if f is 1-1.
2.
In the equation, replace the symbol f (x) with y.
3.
Interchange the roles of x and y and solve equation for y.
4.
Replace y with the symbol f –1.
5.
Verify that f and f –1 are inverses of each other
Show that
f (f -1(x)) = x = f -1(f (x)) and that the domain
of f equals the range of f –1, and the range of f equals the
domain of f –1.
Graphs of f and f –1
Section 2.8
Distance and Midpoint Formulas; Circles
B
A
The distance between any two points A( x1 , y1 ) and B( x2 , y2 ) is:
d(A, B) =
x
2
 x1    y2  y1 
2
2
And the midpoint of the line segment from A to B is
 x1  x2 y1  y2 
,


2
2 

Find the distance and midpoint of the line segment joining the two
points (-5, -3) and (9, 3)
Circle Definition: A circle is the set of all points in a plane that are
equidistant from a fixed point called the center. The fixed distance
from the circle’s center is called the radius.
The standard form of the equation of
a circle with center (h, k) and radius r
(x – h)2 + (y – k)2 = r2
The General Form of the Equation of a Circle
x2 + y2 + Dx + Ey + F = 0
where D, E, and F are reals