Problem 24 - Applied Calculus

Question

(a) Which (if any) of the functions in the following table could be linear? Find formulas for those functions. (b) Which (if any) of these functions could be exponential? Find formulas for those functions. $$ \begin{array}{r|c|c|c} \hline \multicolumn{1}{r|} {x} & f(x) & g(x) & h(x) \\ \hline-2 & 12 & 16 & 37 \\ -1 & 17 & 24 & 34 \\ 0 & 20 & 36 & 31 \\ 1 & 21 & 54 & 28 \\ 2 & 18 & 81 & 25 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

\( h(x) \) is linear as \( h(x) = -3x + 31 \); \( g(x) \) is exponential as \( g(x) = 16 \times 1.5^{x+2} \).

Step 1: Determine Linearity

To assess whether a function is linear, check if the differences between consecutive function values are constant. This means checking if the function values have a constant difference when the input (x) increases by 1 for a given function.

Step 2: Check f(x) for Linearity

Calculate the differences between consecutive values of \( f(x) \). \( 17 - 12 = 5 \) for \( x = -2 \) to \( x = -1 \)\( 20 - 17 = 3 \) for \( x = -1 \) to \( x = 0 \)\( 21 - 20 = 1 \) for \( x = 0 \) to \( x = 1 \)\( 18 - 21 = -3 \) for \( x = 1 \) to \( x = 2 \)Since differences are not constant, \( f(x) \) is not linear.

Step 3: Check g(x) for Linearity

Calculate the differences between consecutive values of \( g(x) \).\( 24 - 16 = 8 \) for \( x = -2 \) to \( x = -1 \)\( 36 - 24 = 12 \) for \( x = -1 \) to \( x = 0 \)\( 54 - 36 = 18 \) for \( x = 0 \) to \( x = 1 \)\( 81 - 54 = 27 \) for \( x = 1 \) to \( x = 2 \)Differences are not constant, so \( g(x) \) is not linear.

Step 4: Check h(x) for Linearity

Calculate the differences between consecutive values of \( h(x) \).\( 34 - 37 = -3 \) for \( x = -2 \) to \( x = -1 \)\( 31 - 34 = -3 \) for \( x = -1 \) to \( x = 0 \)\( 28 - 31 = -3 \) for \( x = 0 \) to \( x = 1 \)\( 25 - 28 = -3 \) for \( x = 1 \) to \( x = 2 \)Differences are constant, thus \( h(x) \) is linear. The formula is \( h(x) = -3x + 31 \).

Step 5: Determine Exponential Functionality

For a function to be exponential, the ratio of consecutive function values should be constant. This means when evaluating the ratios of the function values at successive \( x \), they should be equal for the whole range.

Step 6: Check f(x) for Exponential Behavior

Calculate the ratio of consecutive values of \( f(x) \):\( \frac{17}{12} \approx 1.42 \), \( \frac{20}{17} \approx 1.18 \), \( \frac{21}{20} = 1.05 \), \( \frac{18}{21} \approx 0.86 \)Ratios are not constant, so \( f(x) \) is not exponential.

Step 7: Check g(x) for Exponential Behavior

Calculate the ratio of consecutive values of \( g(x) \):\( \frac{24}{16} = 1.5 \), \( \frac{36}{24} = 1.5 \), \( \frac{54}{36} = 1.5 \), \( \frac{81}{54} = 1.5 \)Since the ratios are constant, \( g(x) \) is exponential. The formula is \( g(x) = 16 \times 1.5^{x+2} \).

Step 8: Check h(x) for Exponential Behavior

Given that h(x) was determined linear in Step 4, it can't be exponential as well.

Key Concepts

Function Linearity

Understanding whether a function is linear is straightforward if you know what to look for. A linear function creates a straight line when plotted on a graph, and its defining characteristic is a constant rate of change.

To check if a function like the ones from our exercise are linear, we focus on the differences between consecutive outputs of the function, given a steady increase in the input. This approach is called checking for constant differences.**What does constant difference mean?**

• If the difference between each successive output value is the same when the input increases by 1 unit, then the function is linear.
• This constant difference is known as the slope of the line. For example, if you move from one point to the next, moving one unit right and up by always the same value, you have a linear relationship.

In our solution, the function \( h(x) \) shows the necessary constant difference (-3), meaning it decreases steadily as \( x \) increases by 1. Hence, the function's formula \( h(x) = -3x + 31 \) reflects this by combining the slope (-3) with the y-intercept (31), where the line would cross the y-axis.

Exponential Functionality

Exponential functions, unlike linear functions, exhibit a rapid increase or decrease as the input value grows. These functions are powerful in representing situations where changes accelerate, such as population growth or radioactive decay.

The hallmark of an exponential function is a constant ratio of consecutive outputs as the inputs increment by equal steps. This means that when you divide one output by the previous output, you consistently get the same quotient.**Understanding constant ratio**

• In a series of outputs, if the division of any output by its predecessor results in the same number, then the function is exponential.
• This number, the constant ratio, serves as the base of the exponential function.

From the example exercise, the function \( g(x) \) has a consistent ratio of 1.5 between each pair of outputs (e.g., \( \frac{24}{16} = 1.5 \)). Hence, the function grows by a factor of 1.5 as \( x \) increases, resulting in the formula \( g(x) = 16 \times 1.5^{x + 2} \). This equation tells us that \( g(x) \) grows exponentially starting from a base value of 16.

Difference and Ratio Analysis

Utilizing difference and ratio analysis is essential for interpreting both linear and exponential functions and determining which type each function is.

**Step-by-Step Analysis**- **Difference Analysis**:

• Calculate the difference between successive outputs. If consistent, the function is linear, as observed with \( h(x) \).
• Use the difference to find the slope and the formula of the linear function.

- **Ratio Analysis**:

• Calculate the ratio of successive outputs. If constant, the function is exponential, like \( g(x) \) in our example.
• This ratio forms the base of the exponential growth formula, showcasing how quickly values grow or decline.

By combining these analyses, one can not only determine the nature of each function in a set but also derive their mathematical representations. Identifying linearity through difference and exponentiality through ratio ensures accurate interpretation and formation of function formulas.