CHAPTER 3

FUNCTIONS AND GRAPHS

3.1 Basic concept of functions• The most useful mathematical idea for modeling the real world is the

concept of function.• Example: If a rock climber drops a stone from a high cliff, what happens to

the stone? Generally the stone will fall; how far it has fallen at any given moment depends upon how long it has been falling; but it doesn’t tell us exactly when the stone will hit the ground. Of course it has a rule that relates the position of the stone to the time it has fallen. The “rule” for finding the distance in term of time is called a function. To understand this rule of function better, we can make a table of values or draw a graph.

• Why function are important?- Physicist finds the “rule” of function that relates distance fallen to elapsed time, the she can predict when a missile will hit the ground.- Biologist finds the function or “rule” that relates the number of bacteria in a culture to the time, then he can predict the number of bacteria for some future time.- Farmer knows the function or “rule” that relates the yield of apples to the number of trees per acre, then he can decide how many trees per acre to plant, to minimize the yield.

Function all around us• In every physical phenomenon, one quantity always depends on another.• Example:

- your height depends on your age. - the temperature depends on the time. - cost mailing a package depends on its weight.

• “Function” is the term that can describe this dependence of one quantity on another.

• Example:- Height is a function of age.- Temperature is a function of date.- Cost mailing a package is a function of weight.- The area of a circle is a function of its radius- The number of bacteria in a culture is a function of time.- The weight of an astronaut is a function of her elevation.- Temperature of water from faucet is a function of time

Definition of function

• A function f is a rule that assigns to each element x in a set A exactly one element, called f(x), in set B.

• Any relationship which takes one element of a first set and assigns to it one and only one element of a second set.

• The first set is said to be the domain of the function and the second set is the co-domain.

• Each element of the first set is mapped onto its image in the second set.

• The set of all images will be a subset of the co-domain and called the range.

Type of functions and relations

Solution

3.2 Graph of Functions• Type of functions with its general equation and graph.

Guidelines for graphing quadratic functions• Zeros. Factor the function to find all its real zeros;

these are the x-intercepts of the graph.• Test points. Find the y-intercept for the function by

letting x=0.• End behavior. Determine the end behavior of the

function by letting x→±∞.• Extreme points: Find the maximum and minimum

value of the function (if any). • Graph. Plot the intercepts and other points you found

above. Sketch a smooth curve that passes through these points and exhibits the required end behavior.

b.

Guidelines for graphing polynomial functions

• Zeros. Factor the polynomial to find all its real zeros; these are the x-intercepts of the graph.

• Test points. Make a table of values for polynomial. Include test points to determine whether the graph of polynomial lies above or below the x-axis on the intervals determined by the zeros. Include the y-intercept in this table.

• End behavior. Determine the end behavior of the polynomial.

• Graph. Plot the intercepts and other points you found in the table. Sketch a smooth curve that passes through these points and exhibits the required end behavior.

Guidelines for graphing rational functions• Factor. Factor the numerator and denominator.• Intercept. Find the x-intercept by determining the zeros of the

numerator, and the y-intercept from the value of the function at x=0

• Vertical asymptotes. Find the vertical asymptotes by determining the zeros of the denominator, and then see if y→+∞ or y→-∞ on each side of every vertical asymptote by using test values.

• Horizontal asymptote. Find the horizontal asymptote (if any) by dividing both numerator and denominator by highest power of x that appears in the denominator, and then letting x→±∞.

• Sketch the graph. Graph the information provided by the first hour steps. Then plot as many additional points as needed to fill in the rest of the graph of the function.

y

Asymptotes of rational functions

3.3 Combining functions

Algebra of functions

Composition of functions

3.4 One-to-one functions and their inverse

3.5 Modeling with functions• Guidelines for modeling with functions

Express the model in words. Identify the quantity you want to model and express it, in words, as a

function of the other quantities in a problem. Choose the variable. Identify all the variables used

to express the function in Step 1. Assign a symbol, such as x, to one variable and express the function the other variables in terms of this symbol.

Set up the model. Express the function in the language of algebra by writing it as a function of the single

variable chosen in Step 2. Use the model. Use the function to answer the questions

posed in the problem.

Modeling with quadratic functions

Modeling with quadratic functions

Modeling with quadratic functions

Modeling with quadratic functions

Modeling: Fitting Polynomial CurveRainfall and crop yield

Modeling: Fitting Polynomial Curve

Modeling: Fitting Polynomial CurveLength-at-Age Data for fish

Modeling: Fitting Polynomial Curve

Modeling with rational functions

The rabbit population is asymptotically to 3000 as t increases.