Problem 25 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points. $$ \begin{array}{ccccc} x & 1 & 2 & 3 & 4 \\ \hline m(x) & 80 & 61 & 42.9 & 25.61 \end{array} $$

Step-by-Step Solution

Answer

The table represents neither a linear nor exponential function.

Step 1: Check for Linearity

To determine if the table represents a linear function, we look for a constant rate of change in the function values. Calculate the differences between consecutive function values.1. Difference between 61 and 80: \( 61 - 80 = -19 \)2. Difference between 42.9 and 61: \( 42.9 - 61 = -18.1 \)3. Difference between 25.61 and 42.9: \( 25.61 - 42.9 = -17.29 \)Since the differences are not constant, the function is not linear.

Step 2: Check for Exponential Growth or Decay

For an exponential function, the ratio of successive values should be constant. Calculate the ratios between consecutive function values.1. Ratio from 80 to 61: \( \frac{61}{80} = 0.7625 \)2. Ratio from 61 to 42.9: \( \frac{42.9}{61} \approx 0.7049 \)3. Ratio from 42.9 to 25.61: \( \frac{25.61}{42.9} \approx 0.5967 \)These ratios are not constant, indicating that the table does not define a perfect exponential function either.

Step 3: Check if it is Neither

Since neither the differences between function values are constant (as required for linearity) nor the ratios between them are constant (as required for exponential functions), the table does not represent a function that is linear or exponential.

Key Concepts

Linear Functions

Linear functions are one of the most basic types of functions you'll encounter in mathematics.
They graph as straight lines and have a constant rate of change, also known as slope. This means that the difference between consecutive function values remains the same.
To identify a linear function from a data set, simply calculate the differences between the consecutive values. If these differences are identical, you have a linear function.
In mathematical terms, a linear function can often be expressed as \[ f(x) = mx + c \] where \( m \) represents the slope and \( c \) is the y-intercept.

Linear functions are easy to work with and very predictable, which is why they're commonly used in real-life scenarios like budgeting or planning where changes occur at a constant rate.

Exponential Functions

Exponential functions differ significantly from linear functions because they involve a rate of change that is not constant but rather proportional to the value of the function itself.
This means that as the function value increases or decreases, its rate of change also accelerates, resulting in the characteristic curved graph.
For a function to be exponential, the ratio of successive values must remain constant. This is because exponential growth or decay occurs at a consistent percentage rate.
Take the function formula: \[ f(x) = a \, b^x \] where \( a \) is the initial value and \( b \) is the base of the function, representing the growth or decay factor.
Exponential functions model many real-world processes such as population growth, radioactive decay, and compound interest, where growth compounds over time.

Constant Rate of Change

The concept of a constant rate of change is essential in determining number patterns, especially when working with linear functions.
When you have a constant rate of change, it suggests that each step forward in the function results in an equal step in the function's output.
Here's how you find it: calculate the difference between pairs of consecutive values in your data set. If these differences are the same, your function has a constant rate of change.
Practical applications include scenarios like calculating speed, which involves a constant rate of distance covered over time.
Recognizing a constant rate of change helps in identifying predictable trends and allows for easy forecasting and analysis.

Successive Values Ratio

Employing the successive values ratio method enables you to determine if a function is exponential.
Instead of looking at differences like in the linear function, you examine the ratios of consecutive outputs.
To do this, divide each value in your set by the previous one. If these ratios are equivalent, the function is exponential.
This method is useful in scenarios like determining the growth rate of investments or epidemics, where each state's outcome is a multiplier of the previous state.
While it provides a powerful way to identify exponential behavior, any variation in the ratio suggests that the function may not strictly be exponential, as consistent ratios are required for a perfect exponential description.