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11th GradeGeneral Mathematics 

Functions: An IntroductionAlgebra has so far been about dealing with numbers and solving for the unknown. This topic will now enable us to visualize problems and represent numbers with points, lines, and graphs.

Learning Objectives

• Define what functions are and know their difference from equations.• Distinguish functions from relations.• Evaluate functions and solve problems involving them.• Represent real-life situations using functions, including piece-wise functions.

Learn about It!

I. Functions, Function Notations, and EquationsFunctionsIn an equation in two variables x and y, the variable y may be expressed as f(x). This is what you call a function, where f is a function of x.

The function f(x) can be thought of as a “machine.”

Example 1: a furniture machineIf you feed a furniture machine with a tree trunk (input), at the other end comes out a finished product, say a wooden table (output).

Example 2: a blender machineIf f(x) is a blender machine and x is a fruit, placing x into f(x) gets you a fruit shake.

Function NotationsThe equation y = 3x + 1, for example, is a candidate function because when you have a value for x, you can get a corresponding value for y. This is one example of a function notation.

Other function notations include• f(x) = 3(x) + 1 y is written as a function of x, or f(x).• x → 3(x) + 1 The arrow is read as “is mapped to.”• f : x → 3(x) + 1 A longer notation of function mapping• f = {(x, y) | y = 3(x) + 1} The vertical bar is read as “such that.”

Take note that f and x may be replaced by any other letter in order to represent problems better; thus, for example, they could take forms such as A(r) for area of a circle in terms of radius, and v(t) for velocity as a function of time.

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Difference between Functions and EquationsEquation—denotes the equality between two expressions

Examples:3x = 4x − 1y = 3x + 12 + 5 = 6 + 1

Function—denotes a relation between two variables: a set of inputs and a set of outputsExample:

y = 3x + 1Suppose you are given expressions that involve an equal symbol (=), how do you determine

whether those expressions are equations or functions?Note that some functions do not require equations such as tables and graphs. A function

needs not be written as an equation at all, but it can be any relation between two variables such that for every independent variable, there is one value for the dependent variable. An equation is merely one way to show a relation, which will be discussed in the next chapter.

II. Functions vs. RelationsRelation—a set of numbers grouped with each other that may or may not represent a patternA relation is simply a set ordered pairs—pairs of numbers (such as x and y) that are arranged in an orderly manner.There are four kinds of relations in ordered pairs:A. One-to-one correspondence

Each value of x is unique and has a unique value of y associated with it.Examples:

x1234

y2468

x1234

y2468

B. Many-to-one correspondenceEach value of y has more than one possible value of x associated with it.Example:

x y1 22 43 64 8

C. One-to-many correspondenceEach value of x has more than one possible value of y associated with it.Example:

x y1 22 43 64 8

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D. Many-to-many correspondenceMore than one value of x corresponds with more than one value of y.Example:

x y1 22 43 64 8

Function—a special kind of relation that gives a unique value of y for each value of xOne-to-one and many-to-one relations are both considered functions.

Example:Given the set of ordered pairs, {(2, 3), (4, 3), (2, 1)}, can you tell whether it represents a function or a relation?

Now, suppose you are given a graph of a relation. A vertical line test can be used to determine if the graph represents a function. If a vertical line is dropped at any part of the graph, and if that line intersects the graph at exactly one point, then the graph represents a function.

Examples:

Function Not a function

III. Evaluating and Graphing FunctionsFunctions will make more sense if you learn how to evaluate them. Evaluating functions is simply a substitution.Example:Given the function f(x) = 3x + 1, x may take on different values, say at x = 1 and x = −2.Substituting x = 1 into f(x) = 3x + 1,

f(1) = 3(1) + 1f(1) = 4

Substituting x = −2 into f(x) = 3x + 1,

f(−2) = 3(−2) + 1f(−2) = −5

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Evaluating Functions with Introduction to GraphsRemember that when you evaluate functions, you are normally given a specific value to plug into the function equation.Example:Evaluate f(x) = 5 − 2x at integer values of x at [−2, 2].The function is evaluated as follows:

x y

−2 9

−1 7

0 5

1 3

2 1

This set of points can also be plotted on the Cartesian plane.

Connect the points to construct the graph of the given function.

f (x) = 5 – 2x

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Piecewise FunctionsA piecewise function is a special kind of function, where there are pieces of different functions put together.

Example:

Examining the graph, you can observe that it looks like two different graphs placed on the same plane—one is curved and one is straight.

Referring to the graph again, you can also evaluate the function across different values of x and find the corresponding values of y.

Hence, without a graph as reference, a piecewise function can be evaluated by looking at the range where x falls and using the corresponding equation.

IV. Solving Problems Involving FunctionsFunctions are working all around us. From predicting stock performances, designing buildings, running airplanes, to the game you are playing in your mobile gadget, ten to a thousand functions are used, running at the same time.

Example 1:The path of a projectile is dictated by the function h(t) = −3t2 + 16t + 12, where h is the height (in meters) of the object at a given time t (in seconds). Find the position of the object after 2 seconds.

To solve this problem, you just have to substitute t = 2 into the given function.h(2) = −3(2)2 + 16(2) + 12h(2) = −3(4) + 32 + 12h(2) = −12 + 32 + 12h(2) = 32

This means that the projectile is 32 meters high after 2 seconds.

Example 2:An element has a half-life of 20 years. How much of the element remains after 60 years if the amount (in grams) of the element is found by the function h(t) = 120(0.5)t/20?Similar to the previous example, how are you going to solve this problem now?

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Interpreting Graphs and TablesMeanwhile, there are also typical applications of functions applied in a sort of abstract manner.

A. Tabular FormExample:The table below shows a metric conversion from one unit to another. Do you know how function works in this table?

Table 1 Weight Conversion Chart (English-Metric and Metric-English)

It is as simple as looking at the corresponding unit of measurement. When you want to convert 5 ounces (oz) to grams (g), just simply look at the column for grams that corresponds to 5 oz. The answer would be 141.88 g.

B. Graphical FormGiven the following climate graph of Baguio City, can you tell at which month was the change in rainfall most drastic? At which month was the temperature fluctuation greatest?

Fig. 1 A climate graph for Baguio City, Philippines

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Learning Tasks

1. Computing machinesBring out your calculators and examine the more complicated buttons like log, x2, exp, sqrt, and sin. Construct a table of values for each of these functions.

2. Conversion graphUsing the graph provided in Chapter IV, Figure 1, tabulate the values based on the graph for both rainfall and temperature.

3. Mathematical DeductionConsider the following pairs of numbers:

(4, 60), (2, 30), (5, 75), (10, 150), (8,120)

Arrange them in order using the first number. Predict the numbers paired with 1, 3, 6, 7, and 9.

Worked Examples

Questions:1. Given f(x) = 5x + 7, what is f(−3)?

2. Given m n n nn

( ) = + −−

3 7 63 2

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, what is m(−5)?

3. A missile’s trajectory was calculated to have followed h(t) = −4t2 + 30t + 12, where h is the height in kilometers and t is the time in minutes. What is the missile’s difference in position between 3 and 5 minutes?

Answers:1. Given f(−3), this means that we are solving the value of f(x) when x = −3.

Taking the given function,f(x) = 5x + 7f(−3) = 5(−3) + 7f(−3) = −15 + 7f(−3) = −8

2. Given m(−5), this means that we are solving the value of m(n) when n = −5.Taking the given function,

m n n nn

m

m

( ) = + −−

−( ) =−( ) + −( ) −

−( ) −

−( ) = ( ) + −

3 7 63 2

53 5 7 5 6

3 5 2

53 25 3

2

2

55 615 2

5 75 35 617

5 3417

5 2

( ) −−( ) −

−( ) = − −−

−( ) =−

−( ) = −

m

m

m

TipsEvaluating functions is as easy as “plugging in.”

Key PointsIn solving, PEMDAS still applies.

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3. Given h(t) = −4t2+ 30t + 12, we can determine the position of the missile when t = 3 and when t = 5.At t = 3,

h(3) = −4(3)2 + 30(3) + 12h(3) = −36 + 90 + 12h(3) = 66 km

At t = 5,h(5) = −4(5)2 + 30(5) + 12h(5) = −100 + 150 + 12h(5) = 62 km

Their difference in height is then the difference between h(3) and h(5).h(3) − h(5) = 66 − 62 = 4 km.

Wrap-Up

Looking for What to Do

Comparing

Functions vs Equations Input and Output

Functions vs Relations One-to-one or Many-to-One / Vertical Line Test

Evaluating functions

Table Find the value on the same row.

Graph Match x with the corresponding y value.

Equation Substitute the value of x and perform PEMDAS.

Word problems Construct a working function and solve for the unknown.

References

Simmons, George F. (2003) Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry. Eugene, Oregon 97401: Resource Publications

http://www.purplemath.com/modules/fcns.htmhttps://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-linear-equations-functions/

8th-functions-and-function-notation/v/what-is-a-function