Question

Compare the functions below.

Answer 1

The 1st, 4th, and 5th answers are correct

Answer 2

Answer:

The true statements are:

4) As x increases on the interval [0, ∞), the rate of change of f eventually exceeds the rate of change of both g and h.

5) A quantity increasing exponentially eventually exceeds a quantity growing quadratically or linearly.

Step-by-step explanation:

f(x)=3^x+2

g(x)=20x+4

h(x)=2x^2+5x+2

1) Over the interval [2, 3], the average rate of change of g is lower than that of both f and h.

Over the interval [a,b], the average rate of change of a function "j" is:

rj=[j(b)-j(a)]/(b-a); with a=2 and b=3

rj=[j(3)-j(2)]/(3-2)

rj=[j(3)-j(2)]/(1)

rj=j(3)-j(2)

For g(x):

rg=g(3)-g(2)

g(3)=20(3)+4→g(3)=60+4→g(3)=64

g(2)=20(2)+4→g(2)=40+4→g(2)=44

rg=64-44→rg=20

For f(x):

rf=f(3)-f(2)

f(3)=3^3+2→f(3)=27+2→f(3)=29

f(2)=3^2+2→f(2)=9+2→f(2)=11

rf=29-11→rf=18

For h(x):

rh=h(3)-h(2)

h(3)=2(3)^2+5(3)+2→h(3)=2(9)+15+2→h(3)=18+15+2→h(3)=35

h(2)=2(2)^2+5(2)+2→h(2)=2(4)+10+2→h(2)=8+10+2→h(2)=20

rh=35-20→rh=15

Over the interval [2, 3], the average rate of change of g (20) is greater than that of both f (18) and h (15), then the first statement is false.

2) As x increases on the interval [0, ∞), the rate of change of g eventually exceeds the rate of change of both f and h.

False, because of f(x) is an exponential function, the rate of f eventually exceeds the rate of change of both g and h.

 

3) When x=4, the value of f(x) exceeds the values of both g(x) and h(x).

x=4→f(4)=3^4+2=81+2→f(4)=83

x=4→g(4)=20(4)+4=80+4→g(4)=84

x=4→h(4)=2(4)^2+5(4)+2=2(16)+20+2=32+20+2→h(4)=54

When x=4, the value of f(x) (83) exceeds only the value of h(x) (54), then the third statement is false.

4) As x increases on the interval [0, ∞), the rate of change of f eventually exceeds the rate of change of both g and h.

True, because of f(x) is an exponential function.

5) A quantity increasing exponentially eventually exceeds a quantity growing quadratically or linearly.

True.

6) When x=8, the value of h(x) exceeds the values of both f(x) and g(x).

x=8→f(8)=3^8+2=6,561+2→f(8)=6,563

x=8→g(8)=20(8)+4=160+4→g(8)=164

x=8→h(8)=2(8)^2+5(8)+2=2(64)+40+2=128+40+2→h(8)=170

When x=8, the value of h(x) (170) exceeds only the value of g(x) (164), then the sixth statement is false.