Question

Two different plants grow each year at different rates, which are represented by the functions f(x) = 4x and g(x) = 5x + 2. What is the first year the f(x) height is greater than the g(x) height?

a) Year 0
b) Year 1
c) Year 2
d) Year 3

Answer 1

Upon comparing the functions year by year, f(x) = 4x is never greater than g(x) = 5x + 2 during the first three years given as options. Therefore, the height of the plant represented by f(x) is not greater than the height of the plant represented by g(x) in any of the provided option years. (option none of the above)

Step-by-step explanation:

To determine in which year the height of the plant represented by function f(x) is greater than the height of the plant represented by function g(x), we need to find when f(x) > g(x). We set the functions equal to each other and solve for x: 4x > 5x + 2

This simplifies to:

-x > 2

x < -2

Since x represents time in years and we can't have negative time, we need to examine the growth year by year starting from year 0:

Year 0: f(0) = 4(0) = 0, g(0) = 5(0) + 2 = 2

Year 1: f(1) = 4(1) = 4, g(1) = 5(1) + 2 = 7

Year 2: f(2) = 4(2) = 8, g(2) = 5(2) + 2 = 12

Year 3: f(3) = 4(3) = 12, g(3) = 5(3) + 2 = 17

As we can see from the calculations above, function f(x) is never greater than function g(x) in these years. None of the provided options (Year 0, Year 1, Year 2, Year 3) have f(x) greater than g(x).