Problem 489 - Precalculus

Question

Determine whether Table 4.32 could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points. $$\begin{array}{|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} \\ \hline f(x) & {3} & {0.9} & {0.27} & {0.081} \\ \hline\end{array}$$

Step-by-Step Solution

Answer

The table represents an exponential function: \(f(x) = 3 \cdot (0.3)^{x-1}\).

Step 1: Identify Linear Change

First, we check if the function is linear by evaluating the change in the y-values. In this case, each y-value change is not constant: \(3-0.9 eq 0.9-0.27\), and so on. Thus, the function is not linear.

Step 2: Identify Exponential Change

Next, check if the function is exponential by calculating the ratio between consecutive y-values: \( \frac{0.9}{3} = 0.3\), \( \frac{0.27}{0.9} = 0.3\), \( \frac{0.081}{0.27} = 0.3\). Since the ratio is constant, the function is exponential.

Step 3: Find the Exponential Function

An exponential function has the form \( f(x) = ab^x \). We identify \(a = 3\) from the initial value when \( x=1\), and the constant ratio (base of the exponent) \(b = 0.3\). Thus, the function is \(f(x) = 3 \cdot (0.3)^{x-1}\).

Key Concepts

Linear Functions

Linear functions are an essential part of mathematics and are commonly used to describe constant change. A linear function can be recognized by its straight-line representation on a graph. A clear indicator of a linear function is that the difference between consecutive outputs, or y-values, remains constant as x increases. This consistent difference is known as the slope.
For example, if we have a dataset where each y-value decreases by 2 as x increases by 1, then such a relationship would be considered linear. In our exercise, the differences between the consecutive y-values are not the same, which clearly indicates that the function is not linear. This is confirmed when you calculate:

• The change from 3 to 0.9 is -2.1
• The change from 0.9 to 0.27 is -0.63
• The change from 0.27 to 0.081 is -0.189

These are not equal differences, thus further solidifying the conclusion.

Function Tables

Function tables are a practical way to organize and analyze relationships between variables. They help in visually checking whether the function behaves linearly, exponentially, or otherwise by listing values of x and the corresponding f(x).
By examining the outputs (f(x)) for changes and patterns, one can deduce the nature of the function. In our example, as x increases, we notice that f(x) does not change linearly, hinting towards an exponential relationship instead. Looking at the table:

• For x = 1, f(x) = 3
• For x = 2, f(x) = 0.9
• For x = 3, f(x) = 0.27
• For x = 4, f(x) = 0.081

This structured approach simplifies identifying patterns, especially when combined with checking the ratios of changes, as seen in the exercise. Understanding and using function tables can dramatically improve problem-solving efficiency.

Ratio Analysis

Ratio analysis is a powerful technique to determine the type of function you are dealing with. When trying to identify an exponential function, examining the ratio of y-values between consecutive points provides a solid clue. Consistent ratios suggest exponential growth or decay.
In the exercise provided, the constant ratio was 0.3, reinforcing the discovery that the function is exponential. Calculating the ratios for each consecutive point:

• From 3 to 0.9, the ratio is \( \frac{0.9}{3} = 0.3 \)
• From 0.9 to 0.27, the ratio is \( \frac{0.27}{0.9} = 0.3 \)
• From 0.27 to 0.081, the ratio is \( \frac{0.081}{0.27} = 0.3 \)

This consistency in the ratio signifies an exponential relation where the function can be expressed in the form \( f(x) = ab^x \). This analysis allows us to find the constants a and b, which are the initial value and the base of the exponent, creating a simplified form for the function.