1. Core 3
Functions

2. After completing this chapter you should be able to:
Represent a mapping by a diagram, by an equation and by a graph
Understand the terms function, domain and range
Combine two or more functions to make a composite function
Know the difference between a ‘one to one’ and a ‘many to one’
Know the relationship between the graph of a function and its inverse

3. Square the set {-1, 1, -2, 2, x}
Mapping diagram
Graph -1 1 -2 2 x 1 4 x²
Domain
Range
y = x²
This is an example of a many to one function

4. Mapping diagrams and graphs
‘add 4’ on the set {-3, 1, 4, 6,x}
Mapping diagram
Graph -3 1 4 6 x 1 5 9 10
x + 4
Domain
Range
y = x + 4
This is an example of a one to one function

5. A function is a special mapping Every element of the domain maps to exactly ONE element of the range
One to one
function
many to one
function
Not a function

6. You can write a function two different ways
f(x) = 2x OR f : x → 2x + 1
You can evaluate a function as well!
g(3) when g(x) = 2x² + 3
g(2) = 2(3)² + 3 = 21
or find a value of x if g(x) = 53
53 = 2x² + 3
2x² = 50
x² = 25
x = ± 5

7. so to find the range of g(x) = 2x² + 3 sketch the graph
and you can find the range of a function by sketching the graph
so to find the range of g(x) = 2x² + 3 sketch the graph
range of g(x) is g(x) ≥ 3

8. your turn exercise 2A page 14 exercise 2B page 16

9. How to change mappings into functions by changing the domain
Lets look at y = √x. if the domain is all of the real numbers (𝑥∈ℝ) then this is not a function (negative numbers don’t get mapped anywhere) if you restrict the domain to x ≥ 0 then all the elements get mapped . we can write this function as f(x) = √x, with domain {𝑥∈ℝ, x ≥ 0 }

10. find the range of f(x) = 3x – 2, domain {x = 1, 2, 3, 4}
the domain is discrete (integer values) so we draw a mapping diagram 1 2 3 4 1 4 7 10
Range of f(x) = 3x – 2 is {1, 4, 7, 10}
f(x) is one to one

11. find the range of g(x) = x², domain {𝑥∈ℝ, -5≤ x ≤ 5}
the domain is continuous so we plot a graph
range of g(x) is 0 ≤ g(x) ≤ 25
g(x) is many to one

12. Check out example 5 on page 18 then do exercise 2C

13. Combining two or more functions
you can combine two functions to make a new more complex function
fg(x) means apply g first followed by f
fg(x) is called a composite function
if f(x) = x² and g(x) = x + 1 then
fg(1) = f(1 + 1)
= f(2)
= 2²
= 4
fg(3) = f(3 + 1)
= f(4)
= 4²
= 16
fg(x) = f(x + 1)
= (x + 1) ²

14. = 1 2 2𝑥+4 +4 = 1 2𝑛 𝑥 +4 = 1 𝑛𝑛(𝑥) =mnn(x) =mn²(x)
And now you get the composite function and have to work out how the basic functions were combined (exciting!)
The basic functions are m(x) = 1 𝑥 , n(x) = 2x + 4, p(x) = x² - 2
First lets look at
2 𝑥 +4
This is the same as
2 1 𝑥 +4
= 2m(x) + 4
= nm(x)
And finally
1 4𝑥+12
= 𝑥+4 +4
= 1 2𝑛 𝑥 +4
= 1 𝑛𝑛(𝑥)
=mnn(x)
=mn²(x)
Now lets try
4x² + 16x +14
=(2x + 4) ² - 2
=[n(x)] ² - 2
= pn(x)

15. Exercise 2D page 22

16. And finally Inverse functions
Inverse functions perform the opposite operation to the function.
It takes elements of the range and maps them back into elements of the domain
Inverse functions only exist for one to one functions
The inverse of f(x) is written f-1(x)
f(x) x y
f-1(x)

17. x →square (x²) → multiply by 2 (2x²) → subtract 7 (2x² - 7)
For many straightforward functions all you need is a flow chart to find the inverse
h(x) = 2x² - 7
Order of operations is:
x →square (x²) → multiply by 2 (2x²) → subtract 7 (2x² - 7)
inverse operations are
√( 𝑥+7 2 ) ← square root ← divide by 2 𝑥+7 2 ← add 7 x + 7 ← x
giving h-1(x) = √( 𝑥+7 2 )

18. you can also find the inverse of a function by changing the subject
you can also find the inverse of a function by changing the subject. f(x) = 3 𝑥 −1 let y = f(x) y = 3 𝑥 −1 y(x – 1) = 3 yx = 3 + y x = 3+𝑦 𝑦 therefore f-1(x) = 3+𝑥 𝑥 check f(4) = 1 , f-1(1) = 4 solution is correct
n.b. to state the inverse function replace the y’s with x

19. worth remembering the range of the function is the domain of the inverse function and vice versa

20. further examples can be found on page 24 – 27 now try exercise 2E page 27