1. Functions – Learning Outcomes
Define functions.
Graph polynomial, exponential, logarithmic, and trigonometric functions.
Form composite functions.
Compare functions and find solutions to 𝑓 𝑥 =𝑔(𝑥).
Recognise surjective, injective, and bijective functions.
Find the inverse of a bijective function.

2. Define Function Functions assign a unique output to a given input.
A single input cannot produce multiple outputs – if it does, it is not a function.
Not Function – by Bin im Garten – CC-BY-SA-3.0
Function – by Bin im Garten – CC-BY-SA-3.0

3. Define Function Domain – the set of possible inputs.
Codomain – the set of possible outputs.
Range – the set of actual outputs.
Domain
Codomain
F = {(1, D), (2, C), (3, C)}
Range = {C, D}
(how inputs and outputs are written)

4. Define Functions Which of the following are functions?

5. Define Functions
Functions are often given as a rule rather than as a domain and range.
For a function f which has a domain X and a codomain Y, the function is written as:
𝑓:𝑋↦𝑌: 𝑥 2 , or
𝑓:𝑥↦ 𝑥 2
𝑓 𝑥 = 𝑥 2
, where the rule is to square the input, 𝑥∈𝑋.

6. Define Functions Write the range of the following functions:
𝑓:𝑋↦𝑌: 𝑥 2 , where 𝑋={1, 2, 3, 4}
𝑔:𝑋↦𝑌: 𝑥 2 −8𝑥+3, where 𝑋={3, 6, 7, 8}
ℎ:𝑋↦𝑌:sin⁡(𝑥), where −1≤𝑥≤1∈𝑋, ℝ
𝑘 𝑥 = 𝑥 , where 𝑥∈ℝ

7. Graph Functions
To graph functions, find some inputs and outputs, and plot them as points on a Cartesian plane.
e.g. plot 𝑓 𝑥 = 𝑥 3 +2 𝑥 2 −5𝑥−6 from 𝑥=−3 to 𝑥=2
e.g. plot 𝑔 𝑥 = 2 𝑥 from 𝑥=−1 to 𝑥=3
e.g. plot ℎ 𝑥 =log(x) from 𝑥=1 to 𝑥=5
e.g. plot 𝑘 𝑥 =sin⁡(𝑥) from 𝑥=−180 to 𝑥=180

8. Form Composite Functions
If the output of one function is used as the input of another function, the combination is called a composite function.
If the output of a function 𝑓 is used as the input of a function 𝑔, we write:
𝑔(𝑓 𝑥 ) or 𝑔∘𝑓(𝑥) , spoken “𝑔 of 𝑓 of 𝑥”
e.g. 𝑓 𝑥 =2𝑥+3 and 𝑔 𝑥 = 𝑥 2 +4𝑥
𝑓 1 =2 1 +3=5
𝑔 𝑓 1 = =25+20=45

9. Form Composite Functions
Sometimes we need to generalise composite functions in terms of the original input.
e.g. 𝑓 𝑥 =2𝑥+3 and 𝑔 𝑥 = 𝑥 2 +4𝑥
𝑓∘𝑔 𝑥 =𝑓 𝑥 2 +4𝑥 =2 𝑥 2 +4𝑥 +3
=2 𝑥 2 +8𝑥+3
𝑔∘𝑓 𝑥 =𝑔 2𝑥+3 = 2𝑥 𝑥+3
=4 𝑥 2 +12𝑥+9+8𝑥+12=4 𝑥 2 +20𝑥+21

10. Form Composite Functions
𝑓∘𝑓 𝑥 =𝑓 2𝑥+3 =2 2𝑥+3
=4𝑥+6
𝑔∘𝑔 𝑥 =𝑔 𝑥 2 +4𝑥 = 𝑥 2 +4𝑥 𝑥 2 +4𝑥
= 𝑥 4 +8 𝑥 𝑥 2 +4 𝑥 2 +16𝑥
= 𝑥 4 +8 𝑥 𝑥 2 +16𝑥
Sometimes these would be written 𝑓 2 (𝑥) and 𝑔 2 (𝑥)

11. Form Composite Functions
Let 𝑓 𝑥 =3𝑥+4, 𝑔 𝑥 = 𝑥 2 , ℎ 𝑥 =6−7𝑥
Find:
𝑓∘𝑔 1
𝑔∘ℎ 0
𝑓∘ℎ∘𝑔 6
Express in terms of 𝑥:
𝑓∘𝑔 𝑥
𝑔∘ℎ 𝑥
𝑓∘ℎ∘𝑔(𝑥)

12. Solve Functions Simultaneously
When two functions give the same output for some particular inputs, they intersect.
They may be solved simultaneously in this case, either graphically or algebraically.
e.g. 𝑓 𝑥 =4−𝑥 and 𝑔 𝑥 =𝑥−10
Find the coordinates where the lines 𝑦=𝑓(𝑥) and 𝑦= 𝑔(𝑥) intersect.

13. Solve Functions Simultaneously
Let 𝑓 𝑥 =𝑔(𝑥)
⇒4−𝑥=𝑥−10
⇒2𝑥=14
⇒𝑥=7
𝑓 7 =4− 7 =−3

14. Solve Functions Simultaneously
Find the point(s) of intersection for the following pairs of functions:
𝑓 𝑥 =6−𝑥, 𝑔 𝑥 = 𝑥 2 −6
𝑓 𝑥 = 5 2 𝑥− 9 2 , 𝑔 𝑥 =1−3𝑥
𝑓 𝑥 = 𝑥 2 , 𝑔 𝑥 =𝑥+6
𝑓 𝑥 =2 𝑥 2 −𝑥+5, 𝑔 𝑥 =6𝑥+2
𝑓 𝑥 =2( 2 2𝑥 ), 𝑔 𝑥 =17 2 𝑥 −8

15. Recognise Surjective Functions
The function 𝑓:𝑋↦𝑌 is surjective if for all 𝑦∈𝑌, there exists an 𝑥∈𝑋 such that 𝑓 𝑥 =𝑦
i.e. The function will output every possible 𝑦 value at some stage.
Surjective
Not surjective

16. Recognise Surjective Functions
How to tell if a function is surjective:
Rearrange the rule in terms of 𝑓(𝑥) (i.e. 𝑦).
Note whether each output is valid.
e.g. 𝑓 𝑥 =4𝑥+3⇒𝑦=4𝑥+3
Rearrange: 𝑥= 𝑦−3 4
Every 𝑦 is valid, so the function is surjective.
e.g. 𝑔 𝑥 = 𝑥 2 +1⇒𝑦= 𝑥 2 +1
Rearrange: 𝑥=± 𝑦−1
As 𝑦−1 cannot be negative (inside a square root), not every 𝑦 is valid, and the function is not surjective.

17. Recognise Surjective Functions
Examples:
Linear functions are surjective.
Quadratic functions are not surjective.
More generally, odd degree polynomials (𝑥, 𝑥 3 , 𝑥 5 …) are surjective, even degree polynomials ( 𝑥 2 , 𝑥 4 , 𝑥 6 …) are not surjective.
Exponential functions are not surjective.
Logarithmic functions are surjective.
sin⁡(𝑥) and cos⁡(𝑥) are not surjective, while tan⁡(𝑥) is surjective.

18. Recognise Injective Functions
A function 𝑓:𝑋↦𝑌 is injective if for all 𝑥 1 , 𝑥 2 ∈𝑋, 𝑓 𝑥 1 = 𝑓 𝑥 2 ⇒ 𝑥 1 = 𝑥 2 .
i.e. For each output, there is exactly one input.
Injective
Not injective

19. Recognise Injective Functions
How to tell if a function is injective:
Input 𝑥 1 and 𝑥 2 into the function.
Set 𝑓 𝑥 1 =𝑓 𝑥 2
If the equation simplifies to 𝑥 1 = 𝑥 2 , the function is injective.
e.g. 𝑓 𝑥 =3𝑥−2.
𝑓 𝑥 1 =𝑓 𝑥 2 ⇒3 𝑥 1 −2=3 𝑥 2 −2
⇒3 𝑥 1 =3 𝑥 2
⇒ 𝑥 1 = 𝑥 2
So the function is injective

20. Recognise Injective Functions
Examples:
Linear functions are surjective.
Quadratic functions are not surjective unless the domain is restricted.
Exponential functions are surjective.
Logarithmic functions are surjective.
sin⁡(𝑥), cos⁡(𝑥), and tan⁡(𝑥) are not surjective.

21. Recognise Bijective Functions
A function 𝑓:𝑋↦𝑌 is bijective if it is both surjective and injective.
i.e. to test if a function is bijective, it must be tested for surjectivity and injectivity.

22. Find Inverse of Bijective Function
If 𝑓:𝑋↦𝑌 is a function, it has an inverse 𝑓 −1 :𝑌↦𝑋.
𝑓 −1 is not necessarily a function, e.g:
𝑓={ 1, 𝐷 , 2, 𝐶 , 3, 𝐶 }, then 𝑓 −1 ={ 𝐷, 1 , 𝐶, 2 , 𝐶, 3 }
Function – by Bin im Garten – CC-BY-SA-3.0

23. Find Inverse of Bijective Function
To find an inverse, follow a similar procedure to recognising surjective functions:
𝑓 𝑥 =4𝑥+3⇒𝑦=4𝑥+3
⇒𝑥= 𝑦−3 4
Then, since we typically use 𝑥 for inputs and 𝑦 for outputs, switch 𝑥 and 𝑦:
𝑦= 𝑓 −1 :𝑥↦ 𝑥−3 4 , or
𝑓 −1 𝑥 = 𝑥−3 4

24. Find Inverse of a Bijective Function
If 𝑓 is a bijective function, then its inverse 𝑓 −1 is also a function.
If a bijective function is compounded with its inverse or vice versa, the output will be the same as the input.
e.g. 𝑓 𝑥 =4𝑥+3, 𝑓 −1 𝑥 = 𝑥−3 4
𝑓∘ 𝑓 −1 7 =𝑓 7−3 4 =𝑓 1 =4 1 +3=7
𝑓 −1 ∘𝑓 4 = 𝑓 − = 𝑓 −1 19 = 19−3 4 =4

25. Find Inverse of a Bijective Function
Find the inverse of the following functions and verify them by testing if 𝑓∘ 𝑓 −1 𝑥 =𝑥
𝑓 𝑥 =3𝑥+4
𝑓 𝑥 =5𝑥−1
𝑓 𝑥 = 𝑥
𝑓 𝑥 = 2𝑥+1 𝑥−1