From J. P. Wood, Spring 2012 1

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.

From J. P. Wood, Spring 2012 2

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: Let f (x) = x2 + 3x + 5

Evaluate: f (2)

f (x + 3)

f (-x)

From J. P. Wood, Spring 2012 3

Graphs of Functions: Graph f (x) = 2x and g (x) = 2x + 4

x 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.

From J. P. Wood, Spring 2012 4

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:

From J. P. Wood, Spring 2012 5

Section 2.2

More on Functions and Their Graphs

Increasing, Decreasing and Constant Functions

Intervals of Increase, Decrease, or Constancy

Relative Maxima and Minima

From J. P. Wood, Spring 2012 6

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

From J. P. Wood, Spring 2012 7

2x + 3, x < 1

-x + 4, x > 1

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) =

From J. P. Wood, Spring 2012 8

y2 – y1

x2 – x1

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 = x1 – x2 ≠ 0

Find the slope of the line passing through the following points:

(-3, -1); (-2, 4) (-3, 4); (2, -2)

From J. P. Wood, Spring 2012 9

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:

From J. P. Wood, Spring 2012 10

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 22

y x= − +

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

From J. P. Wood, Spring 2012 11

Using Intercepts to graph Ax + By + C = 0

Graph 4x – 3y – 6 = 0 Find x-intercept

Find y-intercept

Plot and Connect Points

From J. P. Wood, Spring 2012 12

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

From J. P. Wood, Spring 2012 13

Slope is defined as a Rate of Change: y risemx run

Δ= =Δ

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:

2 1

2 1

( ) ( )f x f xx x

−−

Find the average rate of change of

From J. P. Wood, Spring 2012 14

Section 2.5 Transformations of Functions

Graphs of Common Functions:

Constant Identity Absolute Value

Squaring (Quadratic) Square Root Cubic Cube Root

Vertical and Horizontal Shifts:(Let y = f (x) and c > 0)

h(x) = f (x) + c h(x) = f (x + c)

From J. P. Wood, Spring 2012 15

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

From J. P. Wood, Spring 2012 16

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

From J. P. Wood, Spring 2012 17

f (x)

g(x)

f g

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: (x) = , such that g(x) ≠ 0 Domains of Combination Functions are those real numbers common to both functions

From J. P. Wood, Spring 2012 18

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:

From J. P. Wood, Spring 2012 19

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.

From J. P. Wood, Spring 2012 20

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

From J. P. Wood, Spring 2012 21

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

From J. P. Wood, Spring 2012 22

Section 2.8 Distance and Midpoint Formulas; Circles

B A The distance between any two points ),), 2211 yxyx B( and A( is:

d(A, B) = ( ) ( )212

212 yyxx −+−

And the midpoint of the line segment from A to B is

++

2,

22121 yyxx

Find the distance and midpoint of the line segment joining the two points (-5, -3) and (9, 3)

From J. P. Wood, Spring 2012 23

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