What is a Function?

FUNCTIONS REPRESENTED BY DATA

EXAMPLE 1The average monthly precipitation in Bogota, Colombia, from 1973 to 2003, is given in the table below (Intellicast.com) where the months, January to December, are numbered from 1 to 12

Average Precipitation in Bogota

The average precipitation is a function of the month The domain is discrete {1, 2, 3, …., 12} Units of domain and units of rangeP(5) represents the height of the function when m=5

Exercise 1a. Enter the data in the graphing calculator and display a graph

similar to the one shown above.b. From the table calculate P(6) . What is its meaning? c. Find the height of the function when m=10. d. Find the values of m (from the table and from the graph) that satisfy P(m)=2.2

i) Interpret the solution ii) Does your answer in part (i) prevents P(m) from being

a function? Justify your answer

e. Construct a new table with the precipitation as the inputs and months as the outputs.

i) ( From the table ) Is the precipitation a function of the months? Explain.

ii) Use the scatter plot of the data to justify your answer in part (i).

REMARK. The Average Precipitation in Bogota an example of a function with discrete domain. The domain does not contain interval(s) of real numbers. A line graph can be displayed to help with graphs that have discrete domain

Example 2The table below shows the average height, h(t) in meters, of a population of trees up to 40 years old, where t represents the number of years after they were planted.

In this case h(t) is a function of time t. Why? Although only some inputs are given in the table, the input could be any value on the interval [0,40].

The data are depicted below as a scattered plot, as a line graph (a line segment joins consecutive data points), and as a smooth graph (no corners).

Where is this function

increasing? Decreasing?Concave up?

Concave down?

For each graph determine • Domain• Range

• Highest value of h(t)

• Smallest value of h(t)

Intervals

EXERCISE 7Use the graphs of the functions f(x), g(x) to answer the questions below. Take as domain only the values displayed on each window.

a. Where is f(x)increasing?b. Where is f(x)concave up?c Where is f(x) increasing and concave down? e. Where is g(x)increasing and concave down? g. Find the solution to:

i) f(x)>0 ii) g(x)<0 iii) g(x)=-1

EXERCISE 10

Sketch a graph for each of the following situations including the corresponding units.

Indicate whether the domain in consideration is an interval or discrete. Make sure concavities, if

any, are included. Include proper labeling to identify the main features of the graph.

Determine in each case if the function has a largest/smallest value

a) A population of birds start at a large value, decreases to a small value, and then increases to an intermediate value.

d) An egg is placed in a refrigerator. Sketch the temperature of the egg over time (minutes)

f) The bottom of a 10-foot long ladder leaning against a wall is sliding away from the wall. Display the distance between the wall and the bottom of the ladder with respect to time.