Problem 108 - A Graphical Approach to College Algebra

Question

The table below shows the acreage, in millions, of the total of corn and soybeans harvested annually in the United States. In the table, \(x\) represents the year and \(f\) computes the total number of acres for these two crops. The function \(g\) computes the number of acres for corn only. $$\begin{array}{c|c|c|c|c}\boldsymbol{x} & 2009 & 2010 & 2011 & 2012 \\\\\hline \boldsymbol{f}(\boldsymbol{x}) & 164.0 & 166.3 & 167.6 & 172.5 \\\\\hline \boldsymbol{g}(\boldsymbol{x}) & 86.5 & 88.2 &92.3 & 96.4\end{array}$$ (a) Make a table for a function \(h\) that is defined by the equation \(h(x)=f(x)-g(x)\) (b) Interpret what \(h\) computes.

Step-by-Step Solution

Answer

(a) h(x) table: x: 2009 -> 77.5, 2010 -> 78.1, 2011 -> 75.3, 2012 -> 76.1. (b) h(x) computes soybean acreage.

Step 1: Understanding the Functions

We have two functions: \( f(x) \) which represents the total acreage of corn and soybeans, and \( g(x) \) which represents the acreage of corn only. We need to determine \( h(x) \), which is the difference between these two functions. This means \( h(x) = f(x) - g(x) \) and will represent the acreage of soybeans only, since \( h(x) \) is the remainder when corn acreage is subtracted from the total.

Step 2: Calculate Values of h(x)

For each year, we calculate \( h(x) = f(x) - g(x) \). Do this for each year in the table:- **For 2009:** \( h(2009) = 164.0 - 86.5 = 77.5 \)- **For 2010:** \( h(2010) = 166.3 - 88.2 = 78.1 \)- **For 2011:** \( h(2011) = 167.6 - 92.3 = 75.3 \)- **For 2012:** \( h(2012) = 172.5 - 96.4 = 76.1 \)

Step 3: Create the Table for h(x)

Now that we have calculated the values of \( h(x) \), we can create a table that lists the years along with their corresponding \( h(x) \) values:\[\begin{array}{c|c|c|c|c}\boldsymbol{x} & 2009 & 2010 & 2011 & 2012 \hline\boldsymbol{h}(oldsymbol{x}) & 77.5 & 78.1 & 75.3 & 76.1\end{array}\]

Step 4: Interpret h(x)

The function \( h(x) \) represents the soybean acreage harvested annually in the United States for each of the years in the table. Essentially, \( h(x) \) gives us the number of acres dedicated solely to soybeans by subtracting the corn acreage from the total combined acreage of corn and soybeans.

Key Concepts

Function Notation

In algebra, function notation is used to represent the relationship between two quantities through the use of a function. Here, letters like \( f(x) \) and \( g(x) \) are symbols that help us understand these relationships.

• \( f(x) \): This is read as "\( f \) of \( x \)", where \( x \) is the input value. In our context, \( f(x) \) computes the total acreage of corn and soybeans per year.
• \( g(x) \): Similarly, "\( g \) of \( x \)" tells us the specific annual acreage of corn.
• \( h(x) \): This notation is used to identify the remaining function, which we derive by subtracting corn acreage from the total, to find soybean-only acreage: \( h(x) = f(x) - g(x) \).

Using function notation is convenient as it enables concise expression of complex relationships. It's pivotal to understand the significance of the letters and how logical operations, such as subtraction, are applied in functions like \( h(x) \) to gain deeper insights.

Table Interpretation

Tables are invaluable in organizing data for ease of understanding and analysis. In our problem, we use a table to show the relationship between years and the acreage of crops.
- The first row displays the years (\( x \)) as labels.- Subsequent rows illustrate how each function—\( f(x) \) for total acreage, \( g(x) \) for corn, and \( h(x) \) for soybeans—is computed annually.Each function value in the table corresponds to the specific output when the year (\( x \)) is the input. By examining the table:

• For 2009, \( f(2009) \) indicates 164 million acres harvested in total.
• \( g(2009) \) shows that 86.5 million of these acres are dedicated to corn.
• By calculating \( h(2009) = f(2009) - g(2009) = 77.5 \), we know that 77.5 million are for soybeans.

By interpreting the table, one can see the annual acreage trends for each crop. It offers a snapshot view of how the land use splits between corn and soybeans over different years.

Subtraction of Functions

Subtracting functions is a crucial operation that helps in distinguishing specific aspects within datasets. In this case, to find the soybean acreage, we need to subtract the corn acreage function \( g(x) \) from the total function \( f(x) \).
The calculation is straightforward:- For each year, subtract \( g(x) \) from \( f(x) \) to yield \( h(x) \).- For example, the subtraction for the year 2009 is: \[ h(2009) = f(2009) - g(2009) = 164.0 - 86.5 = 77.5 \]This operation clarifies the specific land allocation to soybeans, filtered out from the total.'s showcases how subtraction is used to highlight a particular feature within a dataset. By using subtraction, you can see exactly how much of the acreage is attributable just to soybeans, eliminating data attributed to corn.