Question

# Use the functions shown :

a. Evaluate each function for x=6 , x=8 and x=12

b. When will function h exceed function f and function g ?

Hint:

### a. We will simply put the values of x given in the question in the 3 functions as per the given graph and find the output/ y values.

## The correct answer is: Function h will exceed functions f & g after point (8,6).

### Step-by-step solution:-

a). f(x) = 0.75x

Let x = 6- f(x) = 0.75(6) = 4.5

Let x = 8- f(x) = 0.75(8) = 6

Let x = 12- f(x) = 0.75(12) = 9

∴ We get the points (6,4.5); (8,6) & (12,9) for f(x)

h(x) = 1.25x

Let x = 6- h(x) = 1.25^{6} = 3.81

Let x = 8- h(x) = 1.25^{8} = 6

Let x = 12- h(x) = 1.25^{12} = 14.55

∴ We get the points (6,3.81); (8,6) & (12,14.55) for h(x)

g(x) = 0.09375x2

Let x = 6- g(x) = 0.09375(6)^{2} = 3.375

Let x = 8- g(x) = 0.09375(8)^{2} = 6

Let x = 12- g(x) = 0.09375(12)^{2} = 13.5

∴ We get the points (6,3.375); (8,6) & (12,13.5) for g(x)

b). From the above calculations, we observe that the 3 functions have a common point (point of intersection) i.e. (8,6).

Hence, after point (8,6), function h will exceed functions f & g because function h is an exponential function while function f & g are linear and quadratic functions, respectively and we know that the rate of increase for an exponential function is higher than that in linear or quadratic functions.

Final Answer:-

∴ Function h will exceed functions f & g after point (8,6).

Let x = 6- f(x) = 0.75(6) = 4.5

Let x = 8- f(x) = 0.75(8) = 6

Let x = 12- f(x) = 0.75(12) = 9

h(x) = 1.25x

Let x = 6- h(x) = 1.25

^{6}= 3.81

Let x = 8- h(x) = 1.25

^{8}= 6

Let x = 12- h(x) = 1.25

^{12}= 14.55

∴ We get the points (6,3.81); (8,6) & (12,14.55) for h(x)

g(x) = 0.09375x2

Let x = 6- g(x) = 0.09375(6)

^{2}= 3.375

Let x = 8- g(x) = 0.09375(8)

^{2}= 6

Let x = 12- g(x) = 0.09375(12)

^{2}= 13.5

∴ We get the points (6,3.375); (8,6) & (12,13.5) for g(x)

b). From the above calculations, we observe that the 3 functions have a common point (point of intersection) i.e. (8,6).

Hence, after point (8,6), function h will exceed functions f & g because function h is an exponential function while function f & g are linear and quadratic functions, respectively and we know that the rate of increase for an exponential function is higher than that in linear or quadratic functions.

Final Answer:-

∴ Function h will exceed functions f & g after point (8,6).