“eff” of x Friday, 23 November 2018

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.

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

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

𝑓 𝑥 = 𝑥 for 𝑥≥0 Range 𝑓 𝑥 ≥0

For all x except x=0 Range 𝑓 𝑥 ≥0

𝑓 𝑥 = sin 𝑥 0≤𝑥≤360 1 −1 Range −1≤𝑓 𝑥 ≤1

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

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−2 1≤𝑥<2 = 6−𝑥 2≤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

Example A function is defined as 𝑓(𝑥) =2𝑥 0≤𝑥<4 =12−𝑥 4≤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

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

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 18+𝑎 =13 𝑎=−5

Example State whether or not the function defined below is continuous. 𝑓 𝑥 =4𝑥−1 0≤𝑥≤2 = 9− 𝑥 2 𝑥≥2 𝑓(2) = 4(2)−1 =7 𝑓(2) = 9−22 =9−4 =5  Function is not continuous

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

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

3. A function is defined as 𝑓(𝑥) =3𝑥 0≤𝑥<1 = 3 1≤𝑥<4 =7−𝑥 4≤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

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=

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=