FUNCTIONS ALLPPT com Free Power Point Templates Diagrams

FUNCTIONS ALLPPT. com _ Free Power. Point Templates, Diagrams and Charts

EXERCISE 1 B. 1 5. If : a. evaluate i. G(2) ii. G(0) iii. G i. ii. iii. b. find a value of x where G(x) does not exist From Then So,

EXERCISE 1 B. 1 5. If : c. find G(x+2) in simplest form d. find x if G(x) = -3 From G(x) = -3 So that

EXERCISE 1 B. 1 6. Represents a function. What is the difference in meaning between and is a function. It takes inputs and then produces an output. The notation is saying more specifically that the function is taking an input x and producing the output.

EXERCISE 1 B. 1 7. If the value of photocopier years after purchase is given by euros: a. find and state what means b. find when c. find the original purchase price of the photocopier. a. V(t) = 9, 650 – 860 t V(4) = 9, 650 – 860(4) = 9, 650 – 3440 = 6, 210 So, the value of photocopier 4 years after purchase is 6, 210 euros.

EXERCISE 1 B. 1 7. If the value of photocopier years after purchase is given by euros: a. find and state what means b. find when c. find the original purchase price of the photocopier. b. From V(t) = 9, 650 – 860 t 5, 780 = 9, 650 – 860 t = 3, 870 t = 4. 5 So, the value of photocopier is 5, 780 euros after years 6 months. 4

EXERCISE 1 B. 1 7. If the value of photocopier years after purchase is given by euros: a. find and state what means b. find when c. find the original purchase price of the photocopier. The original of the photocopier is t = 0 V(t) = 9, 650 – 860 t V(0) = 9, 650 – 860(0) = 9, 650 So, the original purchase price of the photocopier is 9, 650 euros. c.

EXERCISE 1 B. 1 8. On the same set of axes draw the graphs of three different functions f(x) such that f(2)=1 and f(5)=3.

EXERCISE 1 B. 2 3. Find the domain and range of each of the following functions : c. f(x)= |3 x-1|+2 Domain is From y = |3 x – 1| + 2 y – 2 = |3 x – 1| h. Consider And So that So So, domain is Range is

EXERCISE 1 B. 2 3. Find the domain and range of each of the following functions : h. From Consider Then Range is

EXERCISE 1 B. 2 3. Find the domain and range of each of the following functions : i. Domain is From So, range is

EXERCISE 1 B. 2 4. Use a graphics calculator to help sketch graphs of the following functions. Find the domain and range of each. i. k. So, domain is and range is

EXERCISE 1 C 2. Given and find. Find also the domain and range of Give So that Domain of 1. In put is 2. The composite The domain of is and

EXERCISE 1 C 2. Given and find. Find also the domain and range of Range of The composite and Consider Range of is

EXERCISE 1 C 2. Given and find. Find also the domain and range of Domain of 1. In put is 2. The composite The domain of is and

EXERCISE 1 C 2. Given and find. Find also the domain and range of Range of The composite From So that ; The range of is and

EXERCISE 1 C 8. a. If ax + b = cx +d for all values of x, show that a = c and b = d Hint : If it is true for all x, it is true for x = 0 and x = 1 case x = 0 from ax + b = cx +d then a)0(+b=c)0(+d So that b = d case x = 1 from ax + b = cx +d because b = d So ax = cx a=c

EXERCISE 1 C 8. b. Given and that for all values of x, deduce that from fog)x (= f)g)x = f)ax+b = 2)ax+b(+3 = 2 ax+2 b+3 Then fog)x (= x x = 2 ax+2 b+3 and ((consider 2 ax = x ( 2 a = 1 2 b+3 = 0 2 b = -3

EXERCISE 1 C 8. c. Is the result in b true if for all x?

EXERCISE 1 D 2. Draw sign diagrams for : o. For x=1 ; -(5 x+2)(3 x-1) = -(5+2)(3 -1) = -14 < 0

EXERCISE 1 D 2. Draw sign diagrams for : o. As the factors are ‘single’ the signs alternate giving:

EXERCISE 1 D 3. Draw sign diagrams for : c. For x=1 ; We put a – sign here. As the factor is squared the signs do no change.

EXERCISE 1 D 4. Draw sign diagrams for : i. is undefined when x=3 For

EXERCISE 1 D 4. Draw sign diagrams for : i. (x+2) and (x-1) is a single factor then the sing alternates So,

EXERCISE 1 D 4. Draw sign diagrams for : u. is undefined when x=-3 , 2

EXERCISE 1 D 4. Draw sign diagrams for : u. 2 x and (x+8) is a single factor then the sing alternates so,

EXERCISE 1 E 1. Solve for x : c. From So that or

EXERCISE 1 E 2. Solve for x : p. So that,

EXERCISE 1 E 2. Solve for x : q. So,

EXERCISE 1 E 2. Solve for x : r. So,

EXERCISE 1 F. 2 2. Solve for x : c. So, x=-1 or 7

EXERCISE 1 F. 2 3. Solve for x : f. If If So,

EXERCISE 1 F. 2 4. Solve for x b.

EXERCISE 1 F. 3 1. Solve for x : i. From Then

EXERCISE 1 F. 3 2. Solve : f. from

EXERCISE 1 F. 3 3. Solve graphically : d. from

EXERCISE 1 F. 3 5. a. Draw the graph of

EXERCISE 1 F. 3 5. b. P, Q and R are factories which are 5, 2 and 3 km away from factory O respectively. A security service wishes to know where it should locate its premises along AB so that the total length of cable to the 4 factories is a minimum.

i. Explain why the total length of cable is given by where x is the position of security service on AB. is is the the So that distance between x x and and -5 -2 0 3

ii. Where should the security service set up to minimize the length of cable to all 4 factories? . What is the minimum length of cable? . From

From the graph, y has minimum value when In conclusion, security service should be located between factory Q and O which require shortest length of cable that is 10 km.

iii. If a fifth factory at S, located 7 km right of O , also requires the security service, where should the security service locate its premises for minimum cable length?

is is is So that the the the distance distance between between x x x and and and -5 -2 0 3 7

Sketch the graphs of From the graph, y has minimum value when x = 0 In conclusion, security service should be located factory O which require shortest length of cable that is 17 km

EXERCISE 1 H 1. For the following functions : i. determine the asymptotes ii. discuss the behavior of the function as it approaches its asymptotes iii. sketch the graph of all points where the function crosses its asymptotes. iv. find the coordinates of all point where the function crosses its asymptotes. a. i. from Notice that at is undefined As the graph approaches the vertical line is a vertical asymptote. Notice that : and We write : as , we say that

We also notice that and This indicates that y = 0 is a horizontal asymptote and we write: As ) from above( As )from below( Thus x = 2 is a vertical asymptote and y = 0 is a horizontal asymptote.

ii. As the graph approaches the vertical line , we say that is a vertical asymptote. We write : as as This indicates that y = 0 is a horizontal asymptote and we write: As ) from above( As )from below( Thus x = 2 is a vertical asymptote and y = 0 is a horizontal asymptote.

iii. iv. From any point of the graph does not passed through the asymptotes.

EXERCISE 1 H 1. For the following functions : i. determine the asymptotes ii. discuss the behavior of the function as it approaches its asymptotes iii. sketch the graph of all points where the function crosses its asymptotes. iv. find the coordinates of all point where the function crosses its asymptotes. b. i. x = -1 is a vertical asymptote. At x = -1 , f)x (is undefined. As As Notice that

So as as Thus x = -1 is a vertical asymptote and y = 2 is a horizontal asymptote. ii. x = -1 is a vertical asymptote. As As y = 2 is a horizontal asymptote. So as as

iii. iv. From any point of the graph does not passed through the asymptotes.

i. x = 0 is a vertical asymptote. At x = 0 , f)x (is undefined. As As Notice that

So as )from above( as )from below( Thus x = 0 is a vertical asymptote and asymptote. ii. x = 0 is a vertical asymptote. As As is a oblique asymptote. So as )from above( as )from below( is a oblique

iii. iv. From any point of the graph does not passed through the asymptotes.

EXERCISE 1 l 1. For each of the following functions i. on the same axes graph , and ii. find using coordinate geometry and i iii. find using variable interchange: a. i. f(x) = 3 x + 1 passes through (0, 1) and (2, 7) passes through (1, 0) and (7, 2) ii. This line has slope So, its equations is i. e. ,

EXERCISE 1 l 1. For each of the following functions i. on the same axes graph , and ii. find using coordinate geometry and i iii. find using variable interchange: a. iii. is y = 3 x + 1, so i. e. , is

EXERCISE 1 l 1. For each of the following functions i. on the same axes graph , and ii. find using coordinate geometry and i iii. find using variable interchange: b. i. passes through (2, 1) and (6, 2) passes through (1, 2) and (2, 6) ii. This line has slope So, its equations is

EXERCISE 1 l 1. For each of the following functions i. on the same axes graph , and ii. find using coordinate geometry and i iii. find using variable interchange: b.

EXERCISE 1 l 2. For each of the following functions i. find ii. sketch and on the same axes iii. Show that , the identity function: a. i. is y = 2 x + 5, so i. e. , is

EXERCISE 1 l 2. For each of the following functions i. find ii. sketch and on the same axes iii. Show that , the identity function: a. ii.

EXERCISE 1 l 2. For each of the following functions i. find ii. sketch and on the same axes iii. Show that , the identity function: a. iii. and

EXERCISE 1 l 2. For each of the following functions i. find ii. sketch and on the same axes iii. Show that , the identity function: b. i. Is i. e. , , so is

EXERCISE 1 l 2. For each of the following functions i. find ii. sketch and on the same axes iii. Show that , the identity function: b. ii.

EXERCISE 1 l 2. For each of the following functions i. find ii. sketch and on the same axes iii. Show that , the identity function: b. iii. and

EXERCISE 1 l 2. For each of the following functions i. find ii. sketch and on the same axes iii. Show that , the identity function: c. i. is i. e. , , so is

EXERCISE 1 l 2. For each of the following functions i. find ii. sketch and on the same axes iii. Show that , the identity function: c. ii.

EXERCISE 1 l 2. For each of the following functions i. find ii. sketch and on the same axes iii. Show that , the identity function: c. iii. and