Two different businesses model their profits over 15 years, where x is the year, f(x) is the profits of a garden shop, and g(x) is the profits of a construction materials business. Use the data to determine which function is exponential, and use the table to justify your answer.
x f(x) g(x)
1995 $69,682.50 $72,429.27
2000 $78,943.50 $79,967.77
2005 $88,204.50 $88,290.88
2006 $90,056.70 $90,056.70
2007 $91,908.90 $91,857.83
2010 $97,465.50 $97,480.27
f(x) is exponential; an exponential function increases more slowly than a linear function.
f(x) is exponential; f(x) increased more overall than g(x).
g(x) is exponential; g(x) has a higher starting value and higher ending value.
g(x) is exponential; an exponential function increases faster than a linear function.
Where is it what is the answer
To determine which function is exponential, we need to analyze the data and compare the rate of increase for each business over the years.
First, let's observe the data points for f(x):
x f(x)
1995 $69,682.50
2000 $78,943.50
2005 $88,204.50
2006 $90,056.70
2007 $91,908.90
2010 $97,465.50
Looking at the data, we can see that the profits for the garden shop (f(x)) are increasing, but not at a constant rate. The increase in profits from 1995 to 2000 is $9,261.00, while the increase from 2000 to 2005 is $9,261.00 as well. However, the increase from 2005 to 2006 is $1,852.20, and from 2006 to 2007 is $1,852.20 as well.
On the other hand, let's also observe the data points for g(x):
x g(x)
1995 $72,429.27
2000 $79,967.77
2005 $88,290.88
2006 $90,056.70
2007 $91,857.83
2010 $97,480.27
For the construction materials business (g(x)), we can see that the profits are also increasing, but at a fairly consistent rate. The increase in profits from 1995 to 2000 is $7,538.50, which is similar to the increase from 2000 to 2005 ($8,323.11), from 2005 to 2006 ($1,765.82), and from 2006 to 2007 ($1,801.13).
Based on this analysis, we can conclude that g(x) is exponential. The consistent rate of increase in profits suggests that it follows an exponential growth pattern. Conversely, f(x) does not show a consistent rate of increase, indicating that it is not exponential.