1. Functions

2. Starter A rule for a function is 𝑓 𝑐 = 1 𝑐+3
Functions
KUS objectives
BAT understand definition of a function as a one to one mapping
BAT recognise odd and even functions
BAT state the domain and range of functions
Starter A rule for a function is 𝑓 𝑐 = 1 𝑐+3
Evaluate f(2), f(-2), f(0) and f(-5)
Which value of f(c) has no solution ?
Evaluate f(2), f(-2), f(0) and f(-5)

3. Notes Domain and Range – mappings
Instead of finding a single value of f(x) imagine that each number in the set of possible x values is Input to the function: - the corresponding outputs can be represented as a mapping as shown
DOMAIN
RANGE
Because each element in the first set is mapped to exactly one output we say this mapping is one to one

4. Notes Domain and Range – mappings
Consider this mapping
DOMAIN
RANGE
Because some elements in the first set are mapped to the same output we say this mapping is many to one
What other types of mappings can we have?
Can you think of any operations that are one to many?

5. Domain −𝟒<𝐱<𝟓 Range −𝟑<𝒚<𝟑
WB1a Domain and Range – graphically
state the domain and range
Domain −𝟒<𝐱<𝟓
Range −𝟑<𝒚<𝟑

6. Sketch a graph to help figure it out
Domain:
The domain is the set of all possible x-values which will make the function "work", and will output real y-values.
Range:
The range is the resulting y-values we get after substituting all the possible x-values
Sketch a graph to help figure it out
Range
Domain

7. Domain −𝟗<𝐱<𝟖 Range 𝟑<𝒚<𝟖
WB1b Domain and Range – graphically
state the domain and range
Domain −𝟗<𝐱<𝟖
Range 𝟑<𝒚<𝟖

8. WB2a Sketch the graph of y = f(x) a) f(x) = 𝑥 2 −4𝑥+3, −1<𝑥≤4
What is the domain of f ? What is the range of f ?
Range −1<𝑦≤8
Domain −1<𝑥≤4

9. WB2b Draw a sketch of the function defined by:
𝑓:𝑥⟼ 2𝑥+1, −2<𝑥<4 13−𝑥, ≤𝑥<10
and state the range of f(x)
Range −3<𝑦≤9
Domain −2<𝑥≤10

10. WB3ab Domain and Range – graphically
Sketch each graph and state its domain and range
𝒚= 𝒙 𝟐 −𝟔𝒙+𝟏𝟏
𝐟 𝒙 = 𝟑𝒙+𝟐 𝒙
𝒙=𝟎
𝒚=𝟑
Domain 𝒙∈𝑹
Domain 𝒙∈𝑹 𝒙≠𝟎
Range 𝒚∈𝑹 𝒚>𝟐
Range 𝒚∈𝑹 𝒚≠𝟑

11. WB3cd Domain and Range – graphically
Sketch each graph and state its domain and range
g 𝒙 = 𝟏 𝒙 𝟐
𝐡 𝒙 = 𝟏 (𝒙−𝟑)(𝒙+𝟐)
𝒙=𝟎
𝒙=𝟑
𝒙=−𝟐
𝒚=𝟎
Domain 𝒙∈𝑹 𝒙≠𝟎
Domain 𝒙∈𝑹 𝒙≠𝟑 𝒙≠−𝟐
Range 𝒚∈𝑹 𝒚>𝟎
Range 𝒚∈𝑹 𝒚≠𝟎

12. WB4 The function h(x) is defined by ℎ 𝑥 = 1 𝑥 + 2, 𝑥 𝜖 𝑅 𝑥≠0
a) Sketch a graph of h(x)
b) Solve these equations: h(x) = h(x) = h(x) = 1
c) Explain why the equation h(x) = 2 has no solution
𝒙=𝟎
𝒚=𝟐
𝒉 𝒙 = 𝟏 𝒙 +𝟐=𝟑
𝒙=𝟏
𝒉 𝒙 = 𝟏 𝒙 +𝟐=𝟒
𝒙= 𝟏 𝟐
𝒉 𝒙 = 𝟏 𝒙 +𝟐=𝟏
𝒙=−𝟏
𝒉 𝒙 = 𝟏 𝒙 +𝟐=𝟐
𝟏 𝒙 =𝟎 has no solution

13. WB5 Draw a sketch of the function defined by 𝑔(𝑡)=3𝑡+2, and state it’s domain and range

14. Notes Problems with one to many

15. Notes Problems with one to many: geogebra file ‘root (x+a)’
To avoid the problem with one to many – the function 𝑦= 𝑥−1 is usually only defined for positive values

16. WB6 f(x) = 7−𝑥 , Sketch the graph of y = f(x)
What is the domain of f ? What is the range of f ?
Range 𝑦≥0
Domain 𝑥≤7

17. One thing to improve is –
KUS objectives BAT understand definition of a function as a one to one mapping
BAT recognise odd and even functions
BAT state the domain and range of functions
self-assess
One thing learned is –
One thing to improve is –

18. END