Question

A. Plot the data for the functions f(x) and g(x) on a grid and connect the points.

x -2 -1 0 1 2 x -2 -1 0 1 2

f(x) 1/9 1/3 1 3 9 g(x) -4 -2 0 2 4


b. Which function could be described as exponential and which as linear? Explain.


c. If the functions continue with the same pattern, will the function values ever be equal? If so, give estimates for the value of x that will make the function values equals. If not, explain why the function values will never be equal.

Answer 1

b. The function f(x) appears exponential because its graph approaches but does not cross the negative x-axis, while growing at a faster and faster rate to the right (or precisely: as x increases by 1, the value gets multiplied by the same constant, 3.) The function g(x) is linear since g(x) increases by the same amount as x increases in steps of one unit.

c. The graph appears to show that the functions do not intersect, so the function values will not be equal. The function f is already above the function g and it is growing at a faster rate, so they cannot ever be equal.

Explanation:

used the answer above just changed a few words and all

Answer 2

a) see the plots below

b) f(x) is exponential; g(x) is linear (see below for explanation)

c) the function values are never equal

Explanation:

a) a graph of the two function values is attached

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b) Adjacent values of f(x) have a common ratio of 3, so f(x) is exponential (with a base of 3). Adjacent values of g(x) have a common difference of 2, so g(x) is linear (with a slope of 2).

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c) At x ≥ 1, the slope of f(x) is greater than the slope of g(x), and the value of f(x) is greater than the value of g(x), so the curves can never cross for x > 1. Similarly, for x ≤ 0, the slope of f(x) is less than the slope of g(x). Once again, f(0) is greater than g(0), so the curves can never cross.

In the region between x=0 and x=1, f(x) remains greater than g(x). The smallest difference is about 0.73, near x = 0.545, where the slopes of the two functions are equal.