“eff” of x Monday, 25 February 2019

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

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 = 4 2 +2 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

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

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

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

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)=𝑐𝑜𝑠𝑥 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. 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=

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=