Topic 1 Composite Functions

By the end of this topic, you should be able to:

1. Understand and use function notation.
2. Describe and understand a composite function.
3. Work out the inverse of a function and recognize the graphical relationship between a function and its inverse.

Key Words

i. Composite notation

ii. Function

iii. Function notation

iv. Inverse

v. Mapping

Introduction

In Senior One, you learned about number machines and how they transformed different inputs into outputs, depending on the rule(s) followed. This rule is actually a function that relates the given set of inputs to the set of outputs.

Functions are widely used in real life; for example, in programming machines to perform specific tasks and determine compound interests to be earned after a given period of time. Whenever you are describing something that chains processes together, one after the other, you are composing functions.

In this topic, you will understand and use composite functions in solving real-life problems.

Activity 1.0 (Work in groups)

(a) Write down any six tribes of people in Uganda.

(b) Discuss the relationship of the tribes mentioned above.

(c) Represent your responses in (b) above on a diagram.

(d) Comment on your findings.

1.1 Understanding and Using Function Notation

A function notation is a simpler way of describing a function without a lengthy written explanation.

Activity 1.1(a) (Work in groups)

Study the patterns in the equations below.

15-5=10

15- 4 = 11

15-3=12

15-2=13

15-1=14

15-0=15

(a) Describe any pattern you observe in the way the differences change as the integers subtracted from 15 get smaller.

(b) Write down the equation of the pattern described above.

(c) Use the equation in (b) to predict answers of more integers.

(d) Represent your pattern on a diagram. Comment on the diagram.

(e) Write the function notation of your equation from the pattern above.

(f) Identify the domain and range for the function identified in (e).

Activity 1.1(b) (Work in groups)

(a) List down even numbers between 30 and 40.

(b) Study and describe the relation between the even numbers.

(c) Write down the relation mentioned in (b) as a function.

(d) Represent the relation on an arrow diagram. Hence, state the type of mapping on the arrow diagram.

1.2 Describing and Understanding a Composite Function

A composite function is a combination of successive function mappings on the same object.

Activity 1.2(a) (Work in groups)

Researchers have demonstrated that Physical Education is an essential part of healthy living for people of all age groups, especially, those going to school. Many schools in Uganda are now encouraging learners to take part in Physical Education for at least 2 hours per week. 4

(a) Help the Director of Studies of your school to convert 2 hours into minutes and minutes into seconds.

(b) Represent the relations on an arrow diagram. Hence, write down the functions that represent the relations.

1.3 Working out the Inverse of a Function and Recognizing the Graphical Relationship between a Function and Its Inverse

The Inverse of a Function

Activity 1.3(a) (Work in groups)

Juma was driving along Kampala-Gulu Highway. On checking the driving mirror, he realised that he had bypassed his former Mathematics teacher. If you were Juma, explain what you would do.

Activity 1.3(b) (Work in groups)

A fashion designer travelling to United States of America for a fashion show event wants to know what the temperature will be. She is not familiar with the Celsius scale.

a) Write down the formula that will help her relate the degrees Celsius to degrees Fahrenheit.

b) Use that formula to assist her to find out what 75 degrees Fahrenheit is in degrees Celsius.

c) Write down the formula that will be much more convenient to use on her way back to Uganda.

d) Study the formula in (c), and write down the input temperature scale and output temperature scale.

Recognising the Graphical Relationship between a Function and Its inverse

Any function, f, of x can be written as y = f(x). Such a function can be either linear or quadratic. Using the knowledge of drawing graphs, you can represent a function and its inverse on a graph.

Activity 1.3(e) (Work in groups)

Given the function f(x) = 2x + 2:

(a) find the different values of f(x) on the domain -4 ≤ x ≤ 4.

(b) on a graph paper, draw the graph of f(x).

(c) state the inverse function of f(x) = 2x + 2.

(d) on the same graph paper in (b), draw the graph of f'(x).

(e) study the graphs you have drawn on the graph paper and write down your observations.

(f) Compare your responses with those of other groups.

ICT Activity

(a) In groups, use Microsoft Excel or any other software package to illustrate the function f(x) = x2 + 8 and its inverse graphically.

(b) Print out your work and share with other groups.

Topic Summary

In this topic, you have learnt that:

a composite function is a function whose input is itself a function. The composition of a function is done by substituting the variable of one function with another function.

for any two functions f(x) and g(x), fg(x) will be the result of g acting on x, followed by f on g(x).

a function is undefined if the denominator is equal to zero.

for a given function f(x), the inverse, f(x) maps the range back onto the domain.

the graph of f(x) is the image of f(x) after reflection along the line y = x.

Watch the video below to further understand this topic