1. “eff” of x
Friday, 23 November 2018

2. Graphs of Functions
A function is a mapping (which will not include “one to many”) from
Input values Domain to a set of
Output values Range
The mapping maybe “one to one” or it may be “many to one”
A function is not properly defined until the domain is declared –the range will automatically follow.

3. 𝑓(𝑥)=2𝑥+1 for all values of 𝑥
Domain
“one to one” function 1 -½
Range

4. for
domain
“many to one” function 2
Range
(1, -1)

5. 𝑓 𝑥 = 𝑥 for 𝑥≥0
Range 𝑓 𝑥 ≥0

6. For all x except x=0
Range 𝑓 𝑥 ≥0

7. 𝑓 𝑥 = sin 𝑥 ≤𝑥≤360 1 −1
Range −1≤𝑓 𝑥 ≤1

8. Example
Draw a sketch of the graph 𝑦=𝑥2−5𝑥+6, labelling clearly the intersection with the axes. 𝒙 −𝟐 −𝟏 𝟎 𝟏 𝟐 𝟑 𝑦 20 12 6 2 -2 -1 1 2 3 5 10 15 20

9. A function is defined as 𝑓(𝑥) = 𝑥2 0≤𝑥<1 = 3𝑥−2 1≤𝑥<2
Example
A function is defined as
𝑓(𝑥) = 𝑥2 0≤𝑥<1
= 3𝑥− ≤𝑥<2
= 6−𝑥 ≤𝑥<6
Draw the graph of 𝑓(𝑥) on a grid for values of 𝑥 from 0 to 6. 𝑦 3 2 1 1 2 3 4 5 6 7 𝑥

10. Example
A function is defined as
𝑓(𝑥) =2𝑥 0≤𝑥<4
=12−𝑥 ≤𝑥≤12
Draw the graph of 𝑓(𝑥) on a grid for values of 𝑥 from 0 to 12.
Hence find the area enclosed by the graph and the 𝑥-axis. 2 4 6 8 10 12
1 2 ×12×8
Area =
=48

11. A function 𝑓(𝑥) is defined as
𝑓(𝑥) = 2𝑥 0<𝑥<2
= 4 2<𝑥<4
= 12−2𝑥 4<𝑥<5
a) Draw the function defined
b) Calculate the area enclosed by the graph of 𝑦 = 𝑓(𝑥) and the 𝑥-axis 1 2 3 4 5
(b)
Area A =
1/2x2x4 =4
Area B =
2x4 =8
Area C =
½(4+2)x1 =3 A B C
Area = 15

12. for continous functions 18+𝑎 =13 𝑎=−5
Example
Given that the function defined below is continuous find the value of a.
𝑓 𝑥 =2𝑥2+𝑎 0≤𝑥<3
=5𝑥−2 3≤𝑥<5
𝑓 3 =2 3 2+𝑎
=5 3 −2
for continous functions 𝑎 =13
𝑎=−5

13. Example
State whether or not the function defined below is continuous.
𝑓 𝑥 =4𝑥− ≤𝑥≤2
= 9− 𝑥 2 𝑥≥2
𝑓(2) = 4(2)−1 =7
𝑓(2) = 9−22
=9−4 =5
 Function is not continuous

14. Questions
State the range for each of the following functions
(a) 𝑓(𝑥) = 5𝑥−3 for 𝑥≥0
(b) 𝑓(𝑥) = 𝑥 2 +7 for all 𝑥
(c) 𝑓(𝑥) = 1/𝑥 for 𝑥>0
(d) 𝑓(𝑥) = 𝑥 3 for all 𝑥
(e) 𝑓(𝑥)= cos 𝑥 for 0<𝑥<360
Range
Range
Range
Range
Range

15. 2. Sketch the graphs of
𝑓(𝑥) = 𝑥 2 −5
𝑓(𝑥)= 3 𝑥
𝑓(𝑥)=1−6𝑥

16. 3. A function is defined as
𝑓(𝑥) =3𝑥 0≤𝑥<1
= ≤𝑥<4
=7−𝑥 ≤𝑥<7
Draw the graph of 𝑓(𝑥) on a grid for values of 𝑥 from 0 to 6.
Hence what is the range of values for 𝑓(𝑥) 𝑦 3 2 1 1 2 3 4 5 6 7 𝑥
Range

17. 4. Given that the function defined below is continuous find the value of 𝑝.
𝑓(𝑥) =𝑝𝑥2−2𝑥−3 1≤𝑥≤3
=4𝑥+6 𝑥≥3
5. Given that the function below is continuous find the values of 𝑚 and 𝑛.
𝑓(𝑥) =3𝑥−4 0≤𝑥≤2
=8−𝑚𝑥 2≤𝑥≤4
=𝑛−𝑥 𝑥≥4 𝑝= 𝑚= 𝑛=

18. 6. Given that the function below is continuous find the values of 𝑎 and 𝑏.
𝑓(𝑥) =𝑥2−2 0≤𝑥≤1
=𝑎𝑥+𝑏 1≤𝑥≤2
=5−2𝑥 𝑥≥2 𝑎= 𝑏=