Problem 26
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. $$ \begin{array}{|c|c|c|c|c|} \hline x & 1 & 2 & 3 & 4 \\ \hline f(x) & 10 & 20 & 40 & 80 \\ \hline \end{array} $$
Step-by-Step Solution
Verified Answer
The table represents an exponential function: \(f(x) = 5 \cdot 2^x\).
1Step 1: Examine if the function is linear
To determine if the table could represent a linear function, calculate the differences between consecutive values of \(f(x)\). Compute \(20 - 10 = 10\), \(40 - 20 = 20\), and \(80 - 40 = 40\). Since these differences are not constant, the function is not linear.
2Step 2: Examine if the function is exponential
An exponential function has the form \(f(x) = a \cdot b^x\). Calculate the ratio between consecutive values of \(f(x)\). Compute \(\frac{20}{10} = 2\), \(\frac{40}{20} = 2\), \(\frac{80}{40} = 2\). Since the ratios are constant, the function is likely exponential.
3Step 3: Find the exponential function
Using the consistent ratio from Step 2, identify the value of \(b\) as 2. Use the first point (1, 10) to find \(a\). Plug into \(f(x) = a \cdot b^x\): \(10 = a \cdot 2^1\). Solve for \(a\): \(a = 5\). The function is \(f(x) = 5 \cdot 2^x\).
Key Concepts
Linear FunctionsMathematical TablesFunction Identification
Linear Functions
Linear functions are one of the most common types of functions in mathematics. They are characterized by a constant rate of change, which means that when you plot them on a graph, they appear as a straight line. If you have a function of the form \(f(x) = mx + b\), it is considered a linear function. Here, \(m\) represents the slope, which is the rate at which \(f(x)\) changes with respect to \(x\). The \(b\) is the y-intercept, or the point where the line crosses the y-axis.
A quick test to determine if a table of values represents a linear function is to examine the differences between consecutive \(f(x)\) values. These differences should remain constant if the function is linear. Otherwise, the function is not linear. In our initial exercise, the differences \(10, 20, 40\) were not constant, indicating a non-linear pattern.
Understanding linear functions is crucial because they form the basic building blocks of more complex functions. They model many real-world situations where there is a steady and predictable change.
A quick test to determine if a table of values represents a linear function is to examine the differences between consecutive \(f(x)\) values. These differences should remain constant if the function is linear. Otherwise, the function is not linear. In our initial exercise, the differences \(10, 20, 40\) were not constant, indicating a non-linear pattern.
Understanding linear functions is crucial because they form the basic building blocks of more complex functions. They model many real-world situations where there is a steady and predictable change.
Mathematical Tables
Mathematical tables are powerful tools that help you visually analyze and interpret data. They allow you to organize input and output values and provide a framework to identify patterns or relationships between data points.
In a mathematical table, each row typically represents a pair of values for \(x\) and \(f(x)\). By examining these pairs, you can identify whether a function is linear or exponential. You do this by checking for constant differences or ratios among consecutive entries.
For example, with linear functions, the difference between \(f(x)\) values should be constant. In contrast, for exponential functions, the ratios of consecutive \(f(x)\) values should remain the same. This form of analysis is simple yet powerful and applies to numerous applications.
When you understand how to interpret these tables effectively, you gain insights not only into mathematics but also into how such patterns apply to real-world scenarios, such as population growth or interest calculations.
In a mathematical table, each row typically represents a pair of values for \(x\) and \(f(x)\). By examining these pairs, you can identify whether a function is linear or exponential. You do this by checking for constant differences or ratios among consecutive entries.
For example, with linear functions, the difference between \(f(x)\) values should be constant. In contrast, for exponential functions, the ratios of consecutive \(f(x)\) values should remain the same. This form of analysis is simple yet powerful and applies to numerous applications.
When you understand how to interpret these tables effectively, you gain insights not only into mathematics but also into how such patterns apply to real-world scenarios, such as population growth or interest calculations.
Function Identification
Identifying the type of function that a dataset follows is an essential skill in mathematics. Recognizing whether a function is linear, exponential, or neither lets you choose the correct model for analysis or prediction.
To identify these functions, different characteristics need to be examined. For linear functions, you look for constant differences in the output values. For exponential functions, you calculate the ratio of consecutive output values and check for consistency. If neither constant differences nor constant ratios are observed, the function could be more complex or belong to a different class of functions.
In the exercise provided, by looking at the values \(10, 20, 40, 80\), we calculated the ratios to be consistently 2. This consistency suggested an exponential nature, leading us to derive the function \(f(x) = 5 \cdot 2^x\).
Learning how to identify functions allows you to model and solve various mathematical problems and understand natural phenomena modeled by these mathematical equations.
To identify these functions, different characteristics need to be examined. For linear functions, you look for constant differences in the output values. For exponential functions, you calculate the ratio of consecutive output values and check for consistency. If neither constant differences nor constant ratios are observed, the function could be more complex or belong to a different class of functions.
In the exercise provided, by looking at the values \(10, 20, 40, 80\), we calculated the ratios to be consistently 2. This consistency suggested an exponential nature, leading us to derive the function \(f(x) = 5 \cdot 2^x\).
Learning how to identify functions allows you to model and solve various mathematical problems and understand natural phenomena modeled by these mathematical equations.
Other exercises in this chapter
Problem 26
For the following exercises, solve for \(x\) by converting the logarithmic equation to exponential form. \(\log _{3}(x)=2\)
View solution Problem 26
For the following exercises, graph the transformation of \(f(x)=2^{x}\). Give the horizontal asymptote, the domain, and the range. \(f(x)=2^{-x}\)
View solution Problem 27
For the following exercises, refer to \(\underline{\text { Table }} 7 .\) $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hl
View solution Problem 27
Prove that \(b^{x}=e^{x \ln (b)}\) for positive \(b \neq 1\).
View solution