1. Material Taken From: Mathematics for the international student Mathematical Studies SL Mal Coad, Glen Whiffen, John Owen, Robert Haese, Sandra Haese and Mark Bruce Haese and Haese Publications, AND Mathematical Studies Standard Level Peter Blythe, Jim Fensom, Jane Forrest and Paula Waldman de Tokman Oxford University Press, 2012

2. Functions We will be exploring primarily three types of functions:
Mathematical models that link two variables are called functions.
A function is a relationship between two sets: a first set and a second set. Each element ‘x’ of the first set is related to one and only one element ‘y’ of the second set
We will be exploring primarily three types of functions:
Linear
Quadratic
Exponential

3. Practice
Thelonius and Miles play on the basketball team. Clifford plays on the baseball team and Sonny plays on the tennis team.
The set of athletes A = {Thelonius, Miles, Clifford, Sonny}
The set of team B = {Basketball, Baseball, Tennis}
Decide whether these relationships are functions. Justify your answers.
The relationship between the first set A and the second set B: x is an athlete on team y.
The relationship between the first set B and the second set A: x is the team that has athlete y.

4. Practice
Let A = {1, -1, 0, 2, 4}, B = {1, 0, 4}, and C = {1, 0, 4, 16}. Decide whether these relationships are functions. Justify your answers.
The relationship between the first set A and the second set B: ‘the square of x is y’ or ‘y = x2’.
The relationship between the first set A and the second set C: ‘the square of x is y’ or ‘y = x2’.
The relationship between the first set C and the second set A: ‘the square root of x is y’ or ‘y = 𝑥 ’.

5. Domain and Range
Recall that a function is a relationship between two sets: a first set and a second set. Thus,
The first set is called the domain of the function. The elements of the domain, often thought of as ‘x values’, are the independent variables.
For each value of x’ (input) this is one and only one output. This value is sometimes called the image of ‘x.’ The set of all the images (all the outputs) is called the range of the function. The elements of the range, often though of as ‘y values’, are the dependent variables.

6. Domain = {Inputs} and Range = {Outputs}
Domain and Range
INPUT x
DOMAIN
OUTPUT y
RANGE
equation
Doman and range values are written as sets inside curly brackets.
Domain = {Inputs} and Range = {Outputs}

7. Practice Consider the function y = x2 .
Find the image of (i) x = 1 (ii) x = -2
Write down the domain.
Write down the range.
a) y = 1 and y = 4
Unless otherwise stated
always assume the domain will be the set of real numbers ℝ.
b) The domain is the set of real numbers, ℝ
c) The range is y ≥ 0

8. Practice Consider the function y = 𝟏 𝒙 , x ≠ 0
Find the image of (i) x = 2 (ii) x = -½
Write down the domain.
(i) Decide whether y = 0 is and element of the range. Justify your answer.
(ii) Decide whether y = -5 is and element of the range.
Justify your answer.
a) y = ½ and y = -2
b) The domain is the set of real numbers except 0.
c) (i) No solution – Thus not in range (ii) x = -1/5 – Thus in range

9. Graphing Functions
The graph of a function f is the set of points (x, y) on the Cartesian plane where y is the image of x through the function f.
Different letters can be used to name functions, for example: f, g, h …

10. Graphing Functions Drawing Graphs for the IB:
Draw a table of values to find some points on the graph.
On 2mm graph paper, draw and label the axes with suitable scales.
Plot the points.
Join the points with a straight line or a smooth curve.
Draw, for the IB means draw accurately on graph paper

11. Practice Draw the graph of the function y = -x + 1.
Write down the coordinates of the point where the graph of this function intercepts the x-axis and the y-axis.
Decide whether the point A (200, -199) lies on the graph of this function.
The point B (6, y) lies on the graph of this function. Find y.
b) x-intercept (1,0) y-intercept (0,1)
c) yes
d) y = -5

12. Practice Here is the graph of function f Use this graph to find:
The domain of f
The range of f
The points where the graph
intersects the x and y axes
a) f = { x / x ≤1.5}
c) x is (1.5, 0) and (-1.5, 0)
y is (0, -6)
b) f = { y / y ≥−6}

13. Practice
Sketch a graph, in IB, means give a general shape of the graph. In order to do this:
Draw and label the axes.
Label the points where the graph crosses the x and y axes.
Sketch the graph
Sketch the graph of the function y = 3x – 1

14. Function Notation
y = f (x) means that the image of x through the function f is y. x is the independent variable and y is the dependent variable.
For example, if f (x) = 2x – 5
f (3) represents the image of x = 3. To find the value of f (3) substitute x = 3. Thus f (3) = 2(3) – 5 = 1
f (-1) represents the image of x = -1. To find the value of f (-1) substitute x = -1. Thus f (-1) = 2(-1) – 5 = -7

15. Practice Consider the function f (x) = -x2 + 3x
Find the image of x = -2.
Find f (1)
Show that the point (4, -4) lies on the graph of f
a) -10
b) 2
c) –(4)2 + 3(4) = -4

16. Function as Mathematical Models
Functions can be used to describe real-life situations.
Modeling examples:
Bacterial Growth and Time
Rate of Survival and Dose of Drug
Height of Tide and Latitude
Size of Company and Gross Income
Translate the situation into mathematical language and symbols
Interpret the solution in the context of the problem
Find the solution using mathematics

17. Practice
A rectangular piece of cardboard measures 20 cm by 10 cm. Squares of length x cm are cut from each corner. The remaining card is then folded to make an open box. Write a function to model the volume of the box.
V (x) = (20 – 2x)(10 – 2x)x