1. Functions
3. Notice also that the value of ‘4’ in the range is associated with both 2 and –2 in the domain.
Functions provide relationships between two sets of numbers. Numbers from the ‘domain’ set are ‘mapped’ into a single number in the ‘range’ set.
2. This particular function could be expressed by noting that the function just squares the x: f(x) = x2 -2
0.5 2 -8
0.25 4 64
x: Domain
f(x): Range
f(x)
1. This particular function can be expressed by a rather simple mathematical expression. Can you figure out what it is?

2. Functions in real life You experience functions all the time.
Your home heating gas bill is a function of the average monthly outside temperature bill = f(temp)
Your bank account is a function of the day of the month.
$ = function of Day of month
Your car’s speed is a function of how far you depress the gas pedal. speed = f(pedal depression)
Your math mark is a function of your days of attendance.
mark = f(attendance rate)

3. Graphing Functions
It is often easier to see a pattern in a function by making a graph (a picture of it). A picture is worth a thousand ‘evaluations’.

4. Function Example 1 Your height as a function of time; h(t)
The relationship here is actually
h(t) = 45 – 5t2
where h is height in meters and t is time in seconds if you take a physics course.
40 metres
after 1 second

5. Function Example 2
86 km/h
6 inches

6. Evaluating Functions - Review
To evaluate a function means to find its value for any particular input value x from the domain. Given the function f(x) = 2x – 1 we can evaluate the function for any value of x. A table is useful.
The graph of the function (from the table of values)

7. Solving an Equation
Solving is like evaluating; but backwards. You have likely solved lots of equations with algebra. You have hopefully learned to solve equations like: 7 = 2x – 1. What value of x makes this equation true?
x = 4

8. Functions have only one value
Recall from previous studies that a function has a unique (ie: only one) ‘output’ value for any ‘input’ ‘x’, (ie: number from the domain). It must pass the ‘vertical line’ test when graphed.
But a relationship between sets of numbers where some numbers in the domain can be associated with more than one value in the range is called a relation.

9. Relations
A relationship between sets of numbers where any numbers in the domain can be associated with more than one value in the range is called a ‘relation’.
How would you like to have a car that went two speeds when you pressed the gas pedal (you press the pedal 10 inches and the car goes 40 and backwards 10 km/h?) Or how would you like to be in two or three different places at the same time?

10. Back to Functions
Enough for relations! You will not see them again until you study circles in Grade 11 and when we study other ‘conic sections’ in Grade 12.
We will stick with functions that have one unique value for every element in the domain.

11. Discontinuous Functions
Many functions of real numbers are ‘continuous’, they have no breaks in them! But some functions have breaks in them and are ‘discontinuous’!
Notice the open and solid circles on two particular points. The open indicates that the value of the function at x = 14 is not included on the left part of the function, the solid indicates the value of the function at x = 14 [ie: f(14)], is included on the right part of the curve.

12. Discontinuous Functions - Limits
This function when x approaches 14 from the left gets really close to a limiting value of 22, but it never actually gets down to 22.
At exactly an x value of 14 the function takes a leap instantly to a value of 800!
Fortunately in what we perceive in our ‘classical’ sense of the world around us this doesn’t happen in ‘real’ life. Mathematical models of our world don’t normally take quantum leaps. But they do if you study nuclear physics and those type of sciences.
You will learn lots more of this if you study calculus.
Could be my bank account: at exactly the 14th at 1 second after midnight when pay comes in

13. Piecewise Functions
Sometimes a function can be expressed as two (or more) functions ‘pieced’ together. These are called piecewise functions.
y = 2x + 4 if x<1
y = 6; if x  1

14. One-to-One Functions Some functions are ‘one-to-one’
1. For any particular value of x there is one value of of f(x) or ‘y’ if you plot the function on a graph as y = f(x).
2. Knowing that for any particular ‘x’ there is only one ‘y’, and for any particular ‘y’ there is only one ‘x’ is rather nice!