Problem 26 - College Algebra

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. $$ \begin{array}{|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 10 & 20 & 40 & 80 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The table represents an exponential function: \( f(x) = 5 \cdot 2^x \).

Step 1: Examine the Changes in Function Values

First, check the differences between consecutive \( f(x) \) values. If it follows a constant difference, the function might be linear. If it follows a constant ratio, it might be exponential.\( f(2) - f(1) = 20 - 10 = 10 \)\( f(3) - f(2) = 40 - 20 = 20 \)\( f(4) - f(3) = 80 - 40 = 40 \)The differences are not constant, so it is not linear.

Step 2: Determine If the Function is Exponential

Next, check the ratio between successive function values to determine if it's constant, which indicates an exponential function.\( \frac{f(2)}{f(1)} = \frac{20}{10} = 2 \)\( \frac{f(3)}{f(2)} = \frac{40}{20} = 2 \)\( \frac{f(4)}{f(3)} = \frac{80}{40} = 2 \)The ratio is constant at 2, suggesting the function is exponential.

Step 3: Determine the Exponential Function

Now, find the exponential function. The general form is \( f(x) = ab^x \). Given the ratio \( b = 2 \).Use the point \((1, 10)\) to find \( a \):\( f(1) = a \cdot 2^1 = 10 \)\( 2a = 10 \)\( a = 5 \)Thus, the exponential function is \( f(x) = 5 \cdot 2^x \).

Key Concepts

Linear vs Exponential Functions

To determine whether a sequence of numbers forms a linear or exponential function, we start by understanding the key differences between these two types of functions.
Linear functions involve a constant difference between consecutive values, resulting in a straight line when plotted on a graph. If you spot a sequence where each number is a set amount larger than the previous one, it's likely linear.
Exponential functions, on the other hand, exhibit a constant ratio between successive values. This means that each value is a constant multiplier of the previous one. The graph of an exponential function is a curve that increases (or decreases) rapidly.

• Example of a linear function: 2, 4, 6, 8 (constant difference of 2)
• Example of an exponential function: 2, 4, 8, 16 (constant ratio of 2)

In the given exercise, we compare the differences and ratios of the values to decide if it's linear or exponential. Since the differences between the function values are not constant, we infer that it is not linear. However, the ratios are constant, signaling an exponential function.

Function Determination

To determine the type of function represented by a table of values, we must carefully investigate the changes between these values. This process requires two major checks:
First, examine the differences between consecutive values (known as the first differences) to see if they are constant. If they are, this would suggest a linear function. However, in the exercise, the differences are not constant.
Next, calculate the ratio between successive values (known as the common ratio). A constant ratio indicates an exponential function, as seen in our problem.

• Calculate the difference: if constant, function may be linear.
• Calculate the ratio: if constant, function is exponential.

In this particular case, we note that the common ratio between values is 2, confirming it's exponential. Understanding these steps is crucial to determine whether a set of values form a linear or exponential sequence.

Mathematical Ratios

Mathematical ratios play a significant role in identifying exponential functions. A ratio compares two quantities by division and is a powerful tool in mathematics, especially when analyzing exponential growth or decay.
In the context of functions, we determine whether a sequence is exponential by verifying the ratio between successive terms. If this ratio remains consistent, it highlights a repeated pattern of change, indicative of an exponential relationship.
In the exercise, the constant ratio observed as 2 between each function value allowed us to define the function as exponential.

• Example: Successive function values 20, 40, 80, maintaining a ratio of 2.

The consistent mathematical ratio provides us with the multiplier, in this case "b", used in the exponential function formula \( f(x) = ab^x \). Here, "a" is a constant that can be determined from one of the known points in the sequence, further helping define the specific exponential expression. For the function in question, we established that \( f(x) = 5 \cdot 2^x \). Knowing how to utilize mathematical ratios effectively is vital for analyzing and constructing exponential functions.