1. “eff” of x
Monday, 25 February 2019

2. f(x) which reads “eff of ex” is used to indicate a function of x
Functions
f(x) which reads “eff of ex” is used to indicate a function of x
A function is simply an expression in terms of x.
Examples
𝑓(𝑥) = 2𝑥2−𝑥−3
A quadratic function
𝑓(𝑥) = 4𝑥−5
A linear function
𝑓(𝑥) = 𝑠𝑖𝑛𝑥
A trig function

3. Example
Given 𝑓 𝑥 = 𝑥 2 +2𝑥−5 obtain (i) an expression for 𝑓(𝑎)
(ii) the value of 𝑓(4)
(iii) the solutions of the equation 𝑓(𝑎) = 3.
(i) 𝑓 𝑎 = 𝑎 2 +2𝑎−5
(ii) 𝑓 4 = −5
=16+8−5
=19
(iii) 𝑓 𝑎 =3
𝑎 2 +2𝑎−5=3
𝑎 2 +2𝑎−8=0
𝑎+4 𝑎−2 =0
𝑎=−4
𝑎=2

4. Example
Given 𝑓(𝑥) = 4 – 2𝑥
Work out 𝑓(3)
Solve the equation 𝑓(𝑚) = 7
(i) 𝑓(3) = 4 – 2 3
= 4 –6
=−2
(ii) 𝑓(𝑚) = 7
4−2𝑚 = 7
4−7 =2𝑚
− 3 2 =𝑚

5. 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.

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

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

8. 𝑓 𝑥 = 𝑥 for 𝑥≥0
Range 𝑓 𝑥 ≥0

9. For all x except x=0
Range 𝑓 𝑥 ≥0

10. 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

11. 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

12. A function f(x) is defined as
f(x) = 2x 0 < x < 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

13. 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

14. 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

15. 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)=𝑐𝑜𝑠𝑥 for 0<x<360
Range
Range
Range
Range
Range

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

17. 3. 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
4. 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. 5. 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=