Topicm3b4379636547fba4_1528449000663_0Topic

The domain of the functiondomain of the functiondomain of the function

Levelm3b4379636547fba4_1528449084556_0Level

Third

Core curriculumm3b4379636547fba4_1528449076687_0Core curriculum

V. Functions. Basic level. The student:

4) reads from the graph of the function: the domain, the range, roots, monotonic intervals, intervals in which the function takes values not greater (not smaller) or smaller (not greater) than a given number, greatest and smallest values of the function (if they exist) in the closed interval and argumentsargumentsarguments for which the function takes greatest and smallest values.

Timingm3b4379636547fba4_1528449068082_0Timing

45 minutes

General objectivem3b4379636547fba4_1528449523725_0General objective

Using mathematical objects, interpreting mathematical concepts.

Specific objectivesm3b4379636547fba4_1528449552113_0Specific objectives

1. Identifying the domain of a functionfunctionfunction.

2. Communicating in English, developing basic mathematical, computer and scientific competences, developing learning skills.

Learning outcomesm3b4379636547fba4_1528450430307_0Learning outcomes

The student:

- identifies the domain of a function.

Methodsm3b4379636547fba4_1528449534267_0Methods

1. Discussion.

2. Situational analysis.

Forms of workm3b4379636547fba4_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm3b4379636547fba4_1528450127855_0Introduction

At home, students revise the definition of the function and search for names of sets X and Y occurring in this definition.

Procedurem3b4379636547fba4_1528446435040_0Procedure

Students work individually, using computers. Their task is to identify the domain of the functiondomain of the functiondomain of the function that assigns to the sides of a triangle which are equal to 7 and 8 the length of the third side.

[Geogebra applet]

Students use the inequality of the triangle and conclude that in order to build a triangle out of three line segments: 7, 8 and c, the following conditions must be met:
c > 0, c + 7 > 8, 7 + 8 > c and c + 8 > 7.  Hence:  c > 1 and c < 15.

Therefore the domain of the functiondomain of the functiondomain of the function is the interval (1, 15).

Students consider all rectangles whose perimeter is equal to 16. They mark one of the sides as x, therefore the adjacent side is equal to 8 - x. Students notice that such rectangle exist when the condition x > 0 and 8 - x > 0 is met.

They write down the functionfunctionfunction describing the area of the rectangle depending of the length of the sides.

Therefore, the domain of the function P is the interval (0, 8).

Together students formulate the definition of the domain of the function

Definition of the domain of the function

- The domain of the function is the complete set of possible values for which the function makes numerical sense.m3b4379636547fba4_1527752263647_0- The domain of the function is the complete set of possible values for which the function makes numerical sense.

Discussion – how to identify the domain of the functionfunctionfunction that has a fraction or a root in its formula? Students consider the problem on specific examples.

Example

Students identify the domain of the function $w\left(x\right)=\frac{5}{x+6}$.

They notice that dividing anything by 0 is impossible, therefore: x + 6 ≠ 0.

Therefore the domain of the functiondomain of the functiondomain of the function w is the set of real numbers different than -6.

They write it down as: D = R\{-6}.

Example

Students identify the domain of the function $p\left(x\right)=\sqrt{3-x}$.

They notice that the square root is determined for non‑negative numbers, therefore $3 - x \ge 0$.

Therefore, the domain of the functionfunctionfunction p is the set $D=(- \infty , 3>$.

Together students sum up the consideration.

Conclusion:
- If there is a fraction in the formula of the function, then we eliminate from the domain all numbers for which the value of the expression in the denominator is 0.
- If there is a root in the formula of the function, then the expression under the root of the element must have non‑negative values.

Students use obtained information in the exercises.

Task
Does number 2 belong to the domain of the functiondomain of the functiondomain of the function f?

a) $f\left(x\right)=x-2$

b) $f\left(x\right)=\frac{x-2}{x}$

c) $f\left(x\right)=\left(x+2\right)\left(x-2\right)$

d) $f\left(x\right)=\frac{2}{x^2-4}$

Task
Let’s consider all right, tetragon pyramids, whose sum of all edges is equal to 33. To what interval does the length of the side edge of the pyramid belong?

Task
Identify the domain of the function.

a) $y=x^3-1$

b) $y=\sqrt{2x-6}$

c) $y=\sqrt[3]{x-4}$

d) $y=\sqrt{4-3x}$

Task
Give all natural positive numbers that belong to the domain of the functionfunctionfunction.

a) $y=\sqrt{5-x}$

b) $y=\frac{1}{\sqrt{5-x}}$

Task
Let’s all consider pairs of all positive real numbers x and y whose product is three times greater than the sum. Write number y in relation to x. Identify the domain of this correspondencecorrespondencecorrespondence.

An extra task:
Identify the domain of the function $g\left(x\right)=\frac{3}{x^2-2x+1}$.

Lesson summarym3b4379636547fba4_1528450119332_0Lesson summary

Students do the revision exercises.

Then together they sum‑up the classes, by formulating the conclusions to memorise.

- The domain of the functiondomain of the functiondomain of the function is the complete set of possible values for which the functionfunctionfunction makes numerical sense.

Selected words and expressions used in the lesson plan

argumentsargumentsarguments

correspondencecorrespondencecorrespondence

domain of the functiondomain of the functiondomain of the function

elements that belong to the domain of the functionelements that belong to the domain of the functionelements that belong to the domain of the function

formula of the functionformula of the functionformula of the function

functionfunctionfunction

numerical sense of the expressionnumerical sense of the expressionnumerical sense of the expression