Applications of Algebra

Lesson notes

Curriculum details

Strand

Modelling with Algebra (Strand 1)

Sub-strand

Applications of Algebra (1.2)

Content standard

1.1.2.CS.1 - Demonstrate knowledge and understanding of applying algebraic processes and reasoning involving sequence, functions, and linear programming. 1.1.2.LO.1 Examine, analyse, determine and predict other terms in a pattern/sequence. 1.1.2.LO.2 Distinguish among various types of relations, find the domain and range of, and evaluate functions. 1.1.2.LO.3 Show that a function is injective (into) and/or surjective (onto). Find the inverse, describe the relationship between two variables and establish composite functions. 1.1.2.LO.4 Graph linear and quadratic functions and determine the intercepts. 1.1.2.LO.5 Find graphical and algebraic solutions to a system of three linear equations in three variables and apply them to solve real life problems. 1.1.2.LO.6 Perform algebraic manipulations on polynomial functions and graph polynomial functions. 1.1.2.LO.7 Find the domain, range, and zero of a rational function and state when it is undefined. 1.1.2.LO.8 Identify and describe the order of a matrix, the identity matrix and the zero matrix; find the determinant and perform basic arithmetic operations on 2 by 2 matrices (addition and subtraction).

Indicator

1.1.2.LI.7 - Determine the composite of two given functions.

Suggested placement

Semester 1, Week 11 (Week 11 of the year)

Our suggestion, laid out in curriculum order across three terms of twelve weeks. NaCCA does not fix the week, so follow your school's scheme of learning.

Source document note

The official curriculum document has a fault in this entry, so the curriculum text above is transcribed exactly as printed:

• exemplars - p.105: adjacent duplicated CambriaMath characters were collapsed, but this PDF also maps some equation glyphs to the wrong letter or operator; verify every mathematical expression in this row against the rendered page

Curriculum reference

NaCCA curriculum document, p. 105

Exemplars (from the NaCCA curriculum)

Activity 1: Investigate common properties of function.
Collaborative Learning: Learners work in groups/pairs to investigate addition and multiplication properties of function guided to discover that:
- Addition: If 𝑓 ! and 𝑓 ) are two functions from 𝐴 to 𝐵, then 𝑓 ! + 𝑓 ) is defined as:[𝑓 ! + 𝑓 ) ](𝑥) = 𝑓 ! (𝑥) + 𝑓 ) (𝑥).
- Multiplication: if 𝑓 ! and 𝑓 ) are any two functions from 𝐴 to 𝐵, then 𝑓 ! ∙ 𝑓 ) is defined as [𝑓 ! ∙ 𝑓 ) ] (𝑥) = 𝑓1(𝑥) 𝑓2(𝑥).
- Commutativity of functions addition and/or multiplication.
- Function Equality: Two functions are equal only when they have the same domain, the same codomain and same mapping elements from domain to co-domain.
Activity 2: Composite of two functions.
- Collaborative Learning: Learners work in groups/pairs and construct two or more functions [ℎ(𝑥) 𝑎 𝑓(𝑥)], including scenarios that are relevant and applicable to real life. Then, have them swap with a different pair. The new pair should determine the composite functions. Then, they should swap back and make sure that they were accurate according to the creators)
- Collaborative Learning, Experiential Learning: Learners recognise that if 𝑔 is a function from 𝐴 to 𝐵 and 𝑓 a function from 𝐵 to 𝐶, then the composition of 𝑓 and 𝑔, which is denoted as 𝑓 ∘ 𝑔 is the function from 𝐴 to 𝐵

Activity 2: Investigate properties of composite functions.
Collaborative Learning: Learners work in groups/pairs to investigate some common properties of composite functions. Properties include:
- 𝑓 ∘ 𝑔 ≠ 𝑔 ∘ 𝑓, 𝑒 𝑔(𝑥) = 𝑓(𝑥)
- 𝑓 *! ∘ 𝑓 = 𝑓 *! (𝑓(𝑎)) = 𝑓 *! (𝑏) = 𝑎.
- 𝑓 ∘ 𝑓 *! = 𝑓( 𝑓 *! (𝑏)) = 𝑓(𝑎) = 𝑏.
- If 𝑓 and 𝑔 both are one-to-one functions, then 𝑓 is also one-to-one.
- If 𝑓 and 𝑔 both are onto function, then 𝑓 is also onto.
- If 𝑓 and 𝑓 ∘ 𝑔 both are one-to-one functions, then 𝑔 is also one-to-one.
- If 𝑓 and 𝑓 ∘ 𝑔 are onto, then it is not necessary that 𝑔 is also onto.
- (𝑓 ∘ 𝑔) *! = 𝑔 *! ∘ 𝑓 *!
Teaching and Learning Resources:
- Cut out shapes of different colours and orientation
- Maths Technology learning apps, tools and devices
- Chalkboard illustrations
- Worksheets
- Cut-out geometrical shapes of different colours and orientation
Assessment (1.1.2.AS.7). The document marks these depth-of-knowledge levels for this indicator: Level 3 Strategic reasoning.
Consider the functions 𝑝(𝑥) = 𝑥³ and 𝑞(𝑥) = √𝑥. Find the composite function (𝑞 ∘ 𝑝)(𝑥)