Problem 23 - 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. $$ \begin{array}{|c|c|c|c|c|} \hline x & 1 & 2 & 3 & 4 \\ \hline f(x) & 70 & 40 & 10 & -20 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The table represents a linear function: \(f(x) = -30x + 100\).

Step 1: Determine if the function is linear

To determine if the function is linear, check if the differences between consecutive function values, \(f(x)\), have a constant difference. Calculate the first differences: \( 40 - 70 = -30 \), \( 10 - 40 = -30 \), \( -20 - 10 = -30 \). Since all first differences are constant, the function could be linear.

Step 2: Verify the function is not exponential

For an exponential function, the ratio of consecutive \(f(x)\) values should be constant. Calculate the ratios: \( \frac{40}{70} \approx 0.571 \), \( \frac{10}{40} = 0.25 \), \( \frac{-20}{10} = -2 \). The ratios are not consistent, so the function is not exponential.

Step 3: Conclude the function type is linear

Since the function has a constant first difference and does not have a constant ratio, it is a linear function.

Step 4: Find the linear function equation

Use the slope-intercept form \(y = mx + b\). The slope \(m = -30\) (from Step 1). Use one point, say \((1,70)\), to find \(b\): \(70 = -30 \times 1 + b\) Solve for \(b\): \(b = 100\). Thus, the linear function is \(f(x) = -30x + 100\).

Key Concepts

Function Tables

Function tables are a handy way to organize values of a function. They list input values ( usually called x-values er) and the corresponding output values of the function ( often f(x) or y er). In essence, they show a relationship between two sets of numbers.

• Example: Consider the table with x-values 1, 2, 3, and 4, and corresponding f(x) values 70, 40, 10, and -20.
• The table is used to check whether the function is linear, exponential, or neither by analyzing the relationships between x and f(x).

By examining these function tables, we can apply specific techniques to determine the nature of the function without needing to graph it initially.

Exponential Functions

Exponential functions are identified by a constant ratio between successive outputs, rather than a constant difference. These functions model situations where the rate of change is proportional to the current value, such as in compound interester.

• General Form: An exponential function can be expressed as \( f(x) = a \, b^x \).
• The base \( b \) is a constant ratio between outputs, and \( a \) is the initial value when \( x = 0 \).

In the given function table, we calculated the ratios:

• \( \frac{40}{70} \approx 0.571 \)
• \( \frac{10}{40} = 0.25 \)
• \( \frac{-20}{10} = -2 \)

The inconsistent ratios indicate that this function is not exponential. If the function were exponential, all the calculated ratios would equal the same constant value.

Slope-Intercept Form

The slope-intercept form is a straight-forward way to express linear equations. It's written as: \( y = mx + b \), where:

• \( m \) is the slope, indicating how steep the line is.
• \( b \) is the y-intercept, the value of y when x equals zero.

For a linear function, having a constant first difference means the change in y is consistent for each change in x.In the exercise:

• First differences were calculated to be -30, suggesting a slope \( m = -30 \).
• With a point \((1, 70)\), we solved for the y-intercept: \( 70 = -30 \times 1 + b \) leads to \( b = 100 \).

This gives us the linear equation \( f(x) = -30x + 100 \), ensuring the table can indeed be represented by a linear function.

First Differences

First differences can determine if a function is linear. By subtracting consecutive output values ( f(x)er), you can check for constant differences between outputs. If the differences remain constant, the function is likely linear.

• Example Calculation:
  • \( 40 - 70 = -30 \)
  • \( 10 - 40 = -30 \)
  • \( -20 - 10 = -30 \)

All differences calculated as -30. This uniformity confirms the function within the table is linear. Thus, constant first differences mean linear, while a constant ratio is indicative of an exponential function.