Questions: Olive is comparing the growth rates of p(x)=5^x and q(x)=20x using this graph. Based on the graph, Olive concludes that the growth rate of q(x)=20x is always greater than the growth rate of p(x)=5^x. Where is her mistake?

Olive is comparing the growth rates of p(x)=5^x and q(x)=20x using this graph. Based on the graph, Olive concludes that the growth rate of q(x)=20x is always greater than the growth rate of p(x)=5^x. Where is her mistake?

Transcript text: Olive is comparing the growth rates of $p(x)=5^{x}$ and $q(x)=20 x$ using this graph. Based on the graph, Olive concludes that the growth rate of $q(x)=20 x$ is always greater than the growth rate of $p(x)=5^{x}$. Where is her mistake?

Solution

Solution Steps

Step 1: Understand the Functions

We are comparing two functions: $ p(x) = 5^x $ and $ q(x) = 20x $. The function $ p(x) = 5^x $ is an exponential function, while $ q(x) = 20x $ is a linear function. Exponential functions generally grow faster than linear functions for large values of $ x $.

Step 2: Analyze the Growth Rates

To determine where Olive's mistake is, we need to consider the behavior of both functions as $ x $ increases. For small values of $ x $, especially when $ x $ is negative or close to zero, the linear function $ q(x) = 20x $ may appear to grow faster than the exponential function $ p(x) = 5^x $. However, as $ x $ becomes larger, the exponential function will eventually surpass the linear function in terms of growth rate.

Step 3: Identify Olive's Mistake

Olive's mistake is in her conclusion that the growth rate of $ q(x) = 20x $ is always greater than the growth rate of $ p(x) = 5^x $. This is incorrect because, for large values of $ x $, the exponential function $ p(x) = 5^x $ will grow faster than the linear function $ q(x) = 20x $.

Final Answer

The correct statement is: Olive is only considering a small portion of the domains of the functions. If she went farther to the right on the graph, she would observe that the growth rate of $ p(x) = 5^x $ eventually surpasses the growth rate of $ q(x) = 20x $ for large values of $ x $.

\[ \boxed{\text{Olive is only considering a small portion of the domains of the functions. If she went farther to the right on the graph, she would observe that the growth rate of } p(x)=5^{x} \text{ eventually surpasses the growth rate of } q(x)=20x \text{ for large values of } x.} \]

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