Given the two exponential functions f(x)=3.2(1.05)^2x and g(x)=3.1(1.07)^2x

Question

1. Determine which function grows faster.
2.  Use a graph to support your conclusion in part and sketch both functions.
3. If both graphs represent the population growth of bunnies, compare and contrast the behavior of f(x) and g(x)
4.  Identify region on the graph for which

              I. f(x)>g(x)

             II. g(x)>f(x)

             III. f(x)=g(x)

(when answering pls label the variables (1,2,3.. or a,b,c...) when writing the answers, THANK YOU)

Answer 1

Hi Mia M.,

First look at some points and try to see where these functions go.

1. f(0) = 3.2 g(0) = 3.1

f(1) = 3.528 g(1) = 3.549

f(2) = 3.890 g(2) = 4.063

In between x = 0 and x = 1, g(x) becomes greater than f(x) and therefore grows faster.

2. Go to desmos.com and graph both functions. You will see that g(x) grows faster than f(x).

3. Here, g(x) will have a faster population growth of bunnies than will f(x). At say f(x) = g(x) = 50 bunnies, x = 20.5 time period for g(x), and x = 28.2 time period for f(x). So f(x) takes longer.

4. I. f(x) > g(x) from 0 ≤ x < .841, (if x < 0, for growth this is negative time)

II. g(x) > f(x) for x > .841

III. f(x) = g(x) at x = .841

I hope this helps, Joe.

Answer 2

Hello, Mia,

As Mark has asked in his comment, did you graph the two equations? The graph will provide answers to all the questions. Use a graphing calculator, DESMOS online graphing utility, or make a table of x values and their corresponding y values, then plot them on a graph. Wen using DESMOS, enter the equation with "y," instead of f(x) or g(x).

I'll point out that the first expression has the term "(1.05)^2x" while the second is "(1.07)^2x." Although the coefficients (3.2 and 3.1) are different, you should be able to see which one will escalate more quickly.

I hope this helps, Bob