1. Modulus
Function

2. State the range of 𝐠𝐟(𝐱)
Functions Modulus
KUS objectives
BAT Understand and draw graphs of modulus functions
BAT Find intersections of graphs, including modulus graphs
Starter: 𝒇 𝒙 =𝒙+𝟐, 𝒙∈ℛ and 𝒈 𝒙 = 𝟒 𝒙−𝟑 , 𝒙∈ℛ, 𝒙≠𝟑
Find 𝐟𝐠(𝟒)
Find 𝐠𝐟(−𝟐)
Find 𝐟𝐠(𝐱)
Find 𝐠𝐟(𝐱−𝟐)
Find 𝐟𝐠(𝟐𝐱)
State the range of 𝐠𝐟(𝐱)

3. The modulus function makes values positive. We have two cases:
notes
The modulus function makes values positive.
We have two cases:
i) Modulus of the function
Graph: -4 4 x y

4. The modulus function makes values positive. We have two cases:
notes
The modulus function makes values positive.
We have two cases:
ii) Modulus of x within the function f(x)
Graph: -4 4 x y
What type of function is f(x)?

5. 𝒇𝒈 𝒙 = 𝒙 +𝟑 𝒇𝒈 𝒙 = 𝒙+𝟑 WB1 given that 𝒇 𝒙 = 𝒙 and 𝒈 𝒙 =𝒙+𝟑
Sketch the graphs of the composite functions 𝒇𝒈 𝒙 and 𝒈𝒇 𝒙
Indicating clearly which is which 3
𝒇𝒈 𝒙 = 𝒙 +𝟑 -3 3
𝒇𝒈 𝒙 = 𝒙+𝟑

6. Think
Pair
Share
WB2ab Sketch each pair of graphs on the same axes:
𝑦=3𝑥−6
𝑦=3𝑥+1
𝑦=3 𝑥 −6
𝑦= 3𝑥+1

7. Think
Pair
Share
WB2cd Sketch each pair of graphs on the same axes:
𝑦= 𝑥 2 −4
𝑦=4𝑥−3
𝑦= 𝑥 2 −4
𝑦=4 𝑥 −3

8. 𝑓(𝑥)= (𝑥−4) 2 −6 And 𝑓 𝑥 𝑦= 1 𝑥−3 𝑦= 1 𝑥 −3
Think
Pair
Share
WB2ef Sketch each pair of graphs on the same axes:
𝑦= 1 𝑥−3
𝑓(𝑥)= (𝑥−4) 2 −6
𝑦= 1 𝑥 −3
And 𝑓 𝑥

9. two intersections 1st − 𝑥−4 = 1 2 𝑥+4 𝑥=0 point 0, 4 2nd 𝑥−4= 1 2 𝑥+4
WB 𝑓(𝑥)= 𝑥−4 and 𝑔 𝑥 = 1 2 𝑥+4
sketch the graphs of each function then find their points of intersection
two intersections
1st − 𝑥−4 = 1 2 𝑥+4
𝑥=0 point 0, 4
2nd 𝑥−4= 1 2 𝑥+4
𝑥=16 point 16, 12

10. two intersections 1st 2 −𝑥 −3=− 1 2 𝑥+6 𝑥=−6 point −6, 9
WB 𝑓 𝑥 =2 𝑥 −3 and 𝑔 𝑥 =− 1 2 𝑥+6
sketch the graphs of each function then find their points of intersection
two intersections
1st 2 −𝑥 −3=− 1 2 𝑥+6
𝑥=−6 point −6, 9
2nd 𝑥 −3=− 1 2 𝑥+6
𝑥= 18 5 = point , 4.2

11. Neither of these gives a correct solution – look at the graph
WB 𝑓(𝑥)= 1 2 𝑥−6 and 𝑔(𝑥)= 2𝑥−4
sketch the graphs of each function then find their points of intersection
two intersections
1st − 1 2 𝑥−6 =2𝑥−4
𝑥=4 point 4, 4
2nd − 1 2 𝑥−6 =−(2𝑥−4)
𝑥=− point − 4 3 , 20 3
Note that there are two other possibilities trying to solve algebraically
1 2 𝑥−6=2𝑥− and 𝑥−6=−(2𝑥−4)
Neither of these gives a correct solution – look at the graph

12. Put the negative sign on the ‘easiest’ side 𝑥 2 −6𝑥+8=0
WB6 a) sketch the graph of 𝑦= 𝑥 2 −6𝑥
b) hence, solve the equation 𝑥 2 −6𝑥 =8
b) Four intersections
1st 𝑥 2 −6𝑥=8
𝑥 2 −6𝑥−8=0
𝑥= 6± = 𝟑± 𝟏𝟕
2nd 𝑥 2 −6𝑥=−8
Put the negative sign on the ‘easiest’ side
𝑥 2 −6𝑥+8=0
(𝑥−4)(𝑥−2)=0
𝒙=𝟐 𝒙=𝟒

13. WB 𝑓 𝑥 = 2𝑥−6 , 𝑥 𝜖 𝑅
a) Sketch the graph with equation 𝑦=𝑓 𝑥 showing the coordinates of the points where the graph cuts or meets the axes
b) solve 𝑓 𝑥 =10+𝑥
𝑔 𝑥 = 𝑥 2 −2𝑥+4, 𝑥 𝜖 𝑅, 0≤𝑥<7
find 𝑓𝑔 5
Find the range of 𝑔 𝑥
2𝑥−6 =10+𝑥
2𝑥−6=10+𝑥 gives 𝑥=16
2𝑥−6=−(10+𝑥) gives 𝑥=− 4 3
𝑐) 𝑓𝑔 5 =𝑓 19 = 2(19)−6 =32
𝑑) 𝑔 𝑥 = (𝑥−1) 2 + 3, 𝑔 𝑥 ≥3
𝑔 0 =4 , 𝑔 7 =39 range is 3≤ 𝑔 𝑥 ≤39

14. Hence there is a root in the interval [-4, -3]
WB 𝑓 𝑥 = 𝑥 4 −4𝑥−240
Show that there is a root of f(x) = 0 in the interval [-4, -3]
Find the coordinates of the turning point on graph of y = f(x)
Given that 𝑓 𝑥 =(𝑥−4)( 𝑥 3 +𝑎 𝑥 2 +𝑏𝑥+𝑐) find the values of a,b and c
Sketch the graph of y = f(x)
Hence sketch the graph of y = f(x)
𝑓 −4 =32 𝑓 −3 =−147
Hence there is a root in the interval [-4, -3]
𝑓 ′ (𝑥)=4 𝑥 3 −4
𝑓 ′ (𝑥)=0 when 𝑥=1 gives point 1, −243
c) 𝑓(𝑥)=(𝑥−4)( 𝑥 3 +4 𝑥 2 +16𝑥+60)

15. g is defined by: f x = 𝑥 2 −4𝑥+11, 𝑥≥0
8 marks
12 mins
WB9 Exam Q f is defined by: f x = 𝑥−4 −3, 𝑥∈𝑅
g is defined by: f x = 𝑥 2 −4𝑥+11, 𝑥≥0
Solve f x =3
State the range of g(x)
find g𝐟 −𝟐
plus
a) 𝒙−𝟒 −𝟑=𝟑
𝑥−4=6,
𝑥=10
𝒙−𝟒 =6
minus
𝑥−4=−6,
𝑥=−2
b ) 𝒙 𝟐 −𝟒𝒙+𝟏𝟏= 𝒙−𝟐 𝟐 +𝟕
Range is g(x)≥𝟕
𝐟 −𝟐 = −𝟔 −𝟑 =𝟑
g𝐟 −𝟐 = g 𝟑 = (𝟑) 𝟐 −𝟒 𝟑 +𝟏𝟏=𝟖

16. One thing to improve is –
KUS objectives BAT Understand and draw graphs of modulus functions
BAT Find intersections of graphs, including modulus graphs
self-assess
One thing learned is –
One thing to improve is –

17. END