Determine if the data represented in the table below is linear or exponential. Explain your reasoning. Find a possible function for the data: f (x) 3 125 4 150 180 6 216
Determine if the data represented in the table below is linear or exponential. Explain your reasoning. Find a possible function for the data: f (x) 3 125 4 150 180 6 216
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Determine if the data represented in the table below is linear or exponential.
### Explain your reasoning. Find a possible function for the data:
\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
3 & 125 \\
\hline
4 & 150 \\
\hline
5 & 180 \\
\hline
6 & 216 \\
\hline
\end{array}
\]
**Explanation:**
To determine whether the data is linear or exponential, we examine the differences or ratios between consecutive \( f(x) \) values.
For a linear function, the difference between consecutive \( f(x) \) values should be constant. Let's check:
\[
f(4) - f(3) = 150 - 125 = 25
\]
\[
f(5) - f(4) = 180 - 150 = 30
\]
\[
f(6) - f(5) = 216 - 180 = 36
\]
The differences are not constant (25, 30, 36), so the function is not linear.
For an exponential function, the ratio of consecutive \( f(x) \) values should be constant. Let's check the ratios:
\[
\frac{f(4)}{f(3)} = \frac{150}{125} = 1.2
\]
\[
\frac{f(5)}{f(4)} = \frac{180}{150} = 1.2
\]
\[
\frac{f(6)}{f(5)} = \frac{216}{180} = 1.2
\]
The ratios are constant (1.2), so the function is exponential.
**Possible Function for the Data:**
Since the data indicates an exponential function, we can express \( f(x) \) in the form:
\[ f(x) = a \cdot b^x \]
Given the ratio \( 1.2 \) and knowing \( f(x) = 125 \) when \( x = 3 \):
\[ 125 = a \cdot b^3 \]
Using the ratio \( b = 1.2 \):
\[ 125 = a \cdot (1.2)^3 \]
\[ 125 = a \cdot 1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa55013e1-91aa-48be-9c4b-bf8a0e47de22%2F6c6fc945-ec53-42dc-b34b-795f39be9615%2Fqwibhdq_processed.png&w=3840&q=75)
Transcribed Image Text:### Determine if the data represented in the table below is linear or exponential.
### Explain your reasoning. Find a possible function for the data:
\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
3 & 125 \\
\hline
4 & 150 \\
\hline
5 & 180 \\
\hline
6 & 216 \\
\hline
\end{array}
\]
**Explanation:**
To determine whether the data is linear or exponential, we examine the differences or ratios between consecutive \( f(x) \) values.
For a linear function, the difference between consecutive \( f(x) \) values should be constant. Let's check:
\[
f(4) - f(3) = 150 - 125 = 25
\]
\[
f(5) - f(4) = 180 - 150 = 30
\]
\[
f(6) - f(5) = 216 - 180 = 36
\]
The differences are not constant (25, 30, 36), so the function is not linear.
For an exponential function, the ratio of consecutive \( f(x) \) values should be constant. Let's check the ratios:
\[
\frac{f(4)}{f(3)} = \frac{150}{125} = 1.2
\]
\[
\frac{f(5)}{f(4)} = \frac{180}{150} = 1.2
\]
\[
\frac{f(6)}{f(5)} = \frac{216}{180} = 1.2
\]
The ratios are constant (1.2), so the function is exponential.
**Possible Function for the Data:**
Since the data indicates an exponential function, we can express \( f(x) \) in the form:
\[ f(x) = a \cdot b^x \]
Given the ratio \( 1.2 \) and knowing \( f(x) = 125 \) when \( x = 3 \):
\[ 125 = a \cdot b^3 \]
Using the ratio \( b = 1.2 \):
\[ 125 = a \cdot (1.2)^3 \]
\[ 125 = a \cdot 1.
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