Chapter 3. Functions

3.1 What is a Function

Definition of Function A function f is a rule (or a set of rules) that assigns

to each element x in a set A (called the domain of the function) exactly one

element, called f (x), also known as the value or the image of f at x, in a set

B (the co-domain of the function). Examples: 1. The rule/algorithm that assigns each resident of the U.S. a social security number is a function. (What are its domain and range?) 2. The rule that assigns to each person his or her age (in years) is a function. 3. The set {(1, 2), (2, 1), (3, 3)} is a function. 4. The set {(1, 2), (2, 1), (1, 3)} is not a function. (Why not?)

In the notation y = f (x), x is the independent variable, y is the dependent

variable (or the value/image of f at x) .

Note: The co-domain is not necessary the range of f. The range of f is the

set of all actual values of the function. The co-domain is the (possibly larger) set where all the values (and perhaps some other elements that are not values) of the function come from. An easy example to illustrate the

difference is the function f (x) = 2x, where x is any integer. Since twice of

any integer is another integer, the set of all integer Z can be considered as the co-domain (it is also the domain, by the way). The range is, however, only the subset of all even integers.

Piecewise Defined Functions

Example: The unit step function (or Heaviside function), defined by

≥

<=

cx

cxxuc

,1

,0)( .

Example:

≥

<≤+

<−

=

9),2cos(

94,

4,23

)( 5

2

xx

xxe

xx

xF x

.

Finding the Domain of a Function

Some rules of thumb: Polynomial, exponential, sine and cosine functions have as their domain the set of all real numbers. The domain of rational functions and other trigonometric functions is the set of real numbers EXCEPT the zero(s) of their denominators. In general, the domain of a function is the set of all real numbers for which the function’s expression is defined.

Example: 6)( 2 −−= xxxf

Example: 4

1)(

2 +

−=

xxf

3.2 Graphs of Functions

The graph of a function is the set of all ordered pairs (x, f (x)) for all x in

the domain of f. That is, the graph of f is the set of all points (x, y) such

that y = f (x). Some Basic Functions’ Graphs

[See the document "Graphs You Should Know".]

The Vertical Line Test

A curve in the coordinate plane is the graph of a function if and only if no vertical line intersects the curve more than once.

Example: A circle or ellipse is not the graph of any function

Example: The equation y = | x | defines a function, but the equation

x = | y | does not.

Example: Two families of power functions: y = xn and y = 1 / x

n.

When n = an even positive integer: When n = an odd positive integer: Determine if an Equation Defines a Function (algebraically) Example: Without using the vertical line test, can you tell whether each equation describes a function?

(a) x = y3 + 1

(b) y2 = 4x

2 + 1

3.3 Increasing and Decreasing Functions;

Average Rate of Change Increasing and Decreasing Functions

f is increasing on an interval I if f (x1) < f (x2) whenever x1 < x2 in I.

f is decreasing on an interval I if f (x1) > f (x2) whenever x1 < x2 in I.

Example: The function f (x) = mx + b is increasing on the entire number

line if m > 0. It is decreasing on the entire number line if m < 0.

Example: The function f (x) = x2 is decreasing on the interval (−∞, 0), and

is increasing on the interval (0, ∞). It is neither increasing nor decreasing at 0.

Example: The function f (x) = 1/ x is decreasing on the intervals (−∞, 0)

and (0, ∞). (It is undefined at 0.)

Average Rate of Change

The average rate of change of the function y = f (x) between x = a and x =

b is

Avg. rate of change ab

afbf

x

y

−

−=

∆

∆=

)()(.

Geometrically, the average rate of change is the slope of the secant line

between x = a and x = b on the graph of f , that is the line that passes

through the points (a, f (a)) and (b, f (b)).

Example: Find the average rate of change of the function

13

22)(

2

+

+−=

x

xxxf

(i) Between x = 0 and x = 2, and (ii) between x = 1 and x = 4.

Example: Every linear function, y = mx + b, has a constant average rate of

change, m, on any interval.

On any interval from x1 to x2, the average rate of change is

mxx

xxm

xx

mxmx

xx

bmxbmx

xx

afbf=

−

−=

−

−=

−

+−+=

−

−

12

12

12

12

12

12

12

)()()()()(.

3.4 Transformations of Functions How to easily obtain the graphs of many functions from a single “parent”

graph?

Vertical Shifting

For c > 0:

To graph y = f (x) + c, move the graph of y = f (x) upward c units.

To graph y = f (x) − c, move the graph of y = f (x) downward c units.

Example: Graphs of y = x2 + 1 [top] vs. y = x

2 − 1 [bottom]

Horizontal Shifting

For c > 0:

To graph y = f (x − c), move the graph of y = f (x) right c units.

To graph y = f (x + c), move the graph of y = f (x) left c units.

Example: Graphs of y = (x − 1)2 [top] vs. y = (x + 1)

2 [bottom]

Reflecting

To graph y = − f (x), reflect the graph of y = f (x) across the x-axis.

To graph y = f (−x), reflect the graph of y = f (x) across the y-axis.

Example: Graph of y = −x2

Vertical Scaling (Stretching and Shrinking)

To graph y = c f (x), c > 0:

If c > 1, stretch the graph of y = f (x) vertically by a factor of c.

If 1 > c > 0, shrink the graph of y = f (x) vertically by a factor of c.

Example: Comparing the graphs of y = x2 vs. y = 3x

2

Horizontal Scaling (Stretching and Shrinking)

To graph y = f (cx), c > 0:

If c > 1, shrink the graph of y = f (x) horizontally by a factor of 1 / c.

If 1 > c > 0, stretch the graph of y = f (x) horizontally by a factor

of 1 / c.

Example: Comparing the graphs of y = sin x vs. y = sin (x /2)

And compare both against the graph of y = sin 2x.

Even and Odd Functions

An even function is any function f such that

f (−x) = f (x), for all x in its domain.

The graph of any even function is symmetric with respect to the y-axis. Examples: cos(x), sec(x), any constant function, x2, x4, x6, … , x

−2, x −4, …

An odd function is any function f such that

f (−x) = −f (x), for all x in its domain.

The graph of any even function is symmetric with respect to the origin. Examples: sin(x), tan(x), csc(x), cot(x), x, x3, x5, … , x

−1, x −3, …

Most functions, however, are neither even nor odd. There is one function that is both even and odd. (What is it?)

Arithmetic Combinations of Even and Odd Functions The table below summaries the result of performing the common arithmetic operations on a pair of even and/or odd functions:

Even and Even Odd and Odd Even and Odd

+ / − Even Odd Neither

× / ÷ Even Even Odd

The result above can be extended to arbitrarily many terms. For example, a sum of three or more even functions will again be even. (Care needs to be taken in the cases where 3 or more odd functions forming a product/quotient. For example, a product of 3 odd functions will be odd, but a product of 4 odd functions is even.)

3.5 Quadratic Functions; Maxima and Minima A quadratic function is any function of the form

f (x) = ax2 + bx + c,

where a, b, and c are constants; and a ≠ 0. The graph of every quadratic function in 2 variables is a parabola, and conversely each parabola is the graph of some quadratic function, which

could be obtained from the “parent” graph of y = x2 by the transformations

discussed in the last section. The Standard Equation of a Quadratic Function

Every quadratic function y = ax2 + bx + c can be rewritten (using

completing the square) into the standard form:

y = a(x − h)2 + k

The graph is a parabola whose vertex is at (h, k), and the line x = h is

its axis of symmetry. The parabola opens upwards when a > 0, or

downwards when a < 0. The y-intercept is (0, k), and the x-

intercept(s), if exists, is/are given by the quadratic formula (being

roots of the equation ax2 + bx + c = 0.)

As it turns out, the value of h, the x-coordinate of the vertex as well as the

axis of symmetry, is

a

bh

2

−= .

Example: y = −2x2 + 4x + 1

The standard form of the function is y = −2(x − 1)2 + 3

The graph, shown below, can be obtained by transforming the graph

of y = x2 in 4 steps: (1) Translate horizontally rightward 1 unit. (2)

Stretch vertically by a magnification factor of 2. (3) Reflect it in the x-axis. (4) Translate vertically up 3 units.

Example: (#8) y = −x2 + 10x

Maximum or Minimum Value of a Quadratic Function

The maximum or minimum value (the extreme value) of a quadratic function

f (x) = ax2 + bx + c, a ≠ 0, occurs at its vertex,

a

bhx

2

−== .

If a > 0, then the minimum value is

−=

a

bfhf

2)( .

If a < 0, then the maximum value is

−=

a

bfhf

2)( .

Example: If a ball is thrown vertically upward with a velocity of 96 ft/s,

then its height after t seconds is f (t) = 96t − 16t2 in feet. What is the

maximum height reached by the ball?

3.6 Combining Functions

Algebra Combinations of Functions

Let f and g be functions.

Sum (f + g)(x) = f (x) + g(x)

Difference (f − g)(x) = f (x) − g(x)

Product (f · g)(x) = f (x) g(x)

Quotient )(

)()(

xg

xfx

g

f=

, g(x) ≠ 0

The domain of each combination is the intersection of the domains of both f

and g, that is, where both functions are defined; and that denominator isn’t

zero.

Example: Let 1

)(2 −

=x

xxf and g(x) = 1 − x. Find f + g, f − g, f · g,

f / g and g / f . What is the domain of each?

Composition of Functions

Given two functions f and g, the composite function f ◦ g (also known as the

composition of f and g) is defined by

(f ◦ g)(x) = f ( g(x))

Its domain is the set of real numbers where both g(x) and f ( g(x)) are

defined.

Similarly, 3 or more functions can be composed together:

(f ◦ g ◦ h)(x) = f ( g(h(x)))

Note: In general, (f ◦ g)(x) ≠ (g ◦ f)(x). A composite function acts like an assembly line.

Example: Continue from the previous example, find (f ◦ g)(x) and (g ◦

f)(x). Evaluate each composite at x = 4. Now suppose h(x) = 2x2 − x, find

(f ◦ g ◦ h)(x).

3.7 One-to-One Functions and Their Inverses

One-to-One Functions A function is called a one-to-one function if no two elements in its domain have the same image/value, that is,

f (x1) ≠ f (x2), whenever x1 ≠ x2.

Example: Is y = x2 a one-to-one function? How about y = x

3?

If the graph of the function is known, then whether it is one-to-one can be easily check using the horizontal line test.

The Horizontal Line Test

A function is one-to-one if and only if no horizontal line intersects its graph more than once.

Examples: Determine whether each function is one-to-one.

(a) f (x) = | x − 4 | +2

(b) 2100)( xxf −−=

The Inverse of a Function

Let f be a one-to-one function. Then there exists a function, f −1

, called the

inverse function of f , defined by

f −1

(y) = x if and only if f (x) = y

for every y in the domain of f .

The domain of f −1

is the same as the range of f. Similarly, the range of f −1

is the same as the domain of f. The inverse function, if exists, will also be one-to-one. Therefore, it has an

inverse of its own. What is the inverse function of the inverse, (f −1

)−1

?

Note: The notation f −1

is reserved to denote the inverse function. It is

specifically not the reciprocal of f.

Example: f (x) = x and g(x) = 1 / x are not inverse of each other.

Example: Find the inverse of the function defined by the set {(1, 2), (2, 1), (3, 3)}. Cancellation Properties of Inverse Functions

Suppose f is a one-to-one function with domain A and range B, then:

f −1

(f (x)) = x, for all x in A

f (f −1

(x)) = x, for all x in B

The graph of f −1

is obtained by reflecting the graph of f in the line y = x.

Finding the Inverse Function

Given a one-to-one function y = f (x):

1. Interchange x and y → x = f (y)

2. Solve the equation, if possible, for y in terms of x.

3. The resulting equation is y = f −1

(x).

Example: f (x) = x7 − 42

Example: 7

34)(

+

−=

x

xxf

Modeling with Functions

Guidelines for Modeling with Functions

1. Express the model in words. What’s asked, what are given?

Diagrams might help as well. 2. Choose the variable. Identify all quantities and their relations. 3. Set up the model. Express the desired quantity as a function of

one variable. 4. Use the model. For max/min application, for example.

Example: (#3) A rectangular box has a square base. Its height is half the

width of the base. Find a function that models its volume V in terms of its

width w.

Example: Express the area as a function of its base for a right triangle whose hypotenuse is 5 cm long.

Example: (#33) A rancher wants to build a rectangular pen with an area of 100 m2. Find a function that models the length of fencing required. Then find the pen’s dimensions that require the minimum amount of fencing. Example: An open rectangular box is to be made using an 8 inches by 15 inches piece of cardboard by cutting squares from the corners and folding up sides. Express the volume of the box in terms of one of its side. Example: A water tank has the shape of a circular cylinder of radius 2 ft and

height 6 ft. It is partially filled with water to a depth h, 0 ≤ h ≤ 6. Find a

function that expresses the volume of water contained in the tank in terms of

h.

Example: The Highway Department is planning to build a picnic area along a major highway. It is to be rectangular with an area of 900 square yards and is to be fenced off on the three sides not adjacent to the highway. The cost of fence for the sides perpendicular to highway is $1 per yard and the cost of fence for the side parallel to highway is $3 per yard. Write the cost of this project as a function of the length of the side parallel to highway. Example: A 6-foot tall man is walking away from the base of a street light 12 feet above the ground. Express the length of his shadow as a function of his distance from the base of the light. Example: (#24) A rancher with 750 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. Find a function that models the total area of the four pens. Then find the largest possible total area of the four pens.