Problem 36 - A Graphical Approach to Precalculus with Limits

Question

Use the table to evaluate each expression, if possible. (a) \((f+g)(2)\)(b) \((f-g)(4)\)(c) \((f g)(-2)\)(d) \(\left(\frac{f}{g}\right)(0)\) $$\begin{array}{r|r|r}x & f(x) & g(x) \\\\-2 & -4 & 2 \\\0 & 8 & -1 \\\2 & 5 & 4 \\\4 & 0 & 0\end{array}$$

Step-by-Step Solution

Answer

(a) 9, (b) 0, (c) -8, (d) -8

Step 1: Evaluate \((f+g)(2)\)

The expression \((f+g)(2)\) represents the sum of \(f(2)\) and \(g(2)\). Looking at the table, we find \(f(2) = 5\) and \(g(2) = 4\). Therefore, \((f+g)(2) = f(2) + g(2) = 5 + 4 = 9\).

Step 2: Evaluate \((f-g)(4)\)

The expression \((f-g)(4)\) represents the difference between \(f(4)\) and \(g(4)\). According to the table, \(f(4) = 0\) and \(g(4) = 0\). Therefore, \((f-g)(4) = f(4) - g(4) = 0 - 0 = 0\).

Step 3: Evaluate \((fg)(-2)\)

The expression \((fg)(-2)\) represents the product of \(f(-2)\) and \(g(-2)\). From the table, \(f(-2) = -4\) and \(g(-2) = 2\). Therefore, \((fg)(-2) = f(-2) \cdot g(-2) = -4 \cdot 2 = -8\).

Step 4: Evaluate \(\left(\frac{f}{g}\right)(0)\)

The expression \(\left(\frac{f}{g}\right)(0)\) represents the quotient of \(f(0)\) and \(g(0)\). According to the table, \(f(0) = 8\) and \(g(0) = -1\). Therefore, \(\left(\frac{f}{g}\right)(0) = \frac{f(0)}{g(0)} = \frac{8}{-1} = -8\).

Key Concepts

Function Addition

Function addition involves combining two functions by adding their corresponding outputs. If you have functions \(f(x)\) and \(g(x)\), the sum is noted as \((f+g)(x)\). To find this, simply evaluate each function separately and add the results together.

In our exercise, we calculated \((f+g)(2)\) by finding \(f(2) = 5\) and \(g(2) = 4\). Adding these gives us 9.

Function addition follows these simple steps:

• Identify the x-value to evaluate.
• Calculate the output for both \(f(x)\) and \(g(x)\) at that point.
• Add the results to get the sum.

Remember, the reason we add functions is to see how two processes or phenomena might combine at any given point. It's like figuring out a combined effect.

Function Subtraction

Function subtraction is quite similar to addition, except we subtract one function's output from the other. When you have \(f(x)\) and \(g(x)\), their difference can be represented as \((f-g)(x)\). Look at the results from each function and subtract.

In our example, we evaluated \((f-g)(4)\) by checking \(f(4) = 0\) and \(g(4) = 0\). Subtracting gives 0, meaning both functions equal out at this point.

Here's how to do it:

• Select the x-value you are interested in.
• Find the value of both \(f(x)\) and \(g(x)\).
• Subtract the value of \(g(x)\) from \(f(x)\).

This operation helps us understand differences or changes between two functions at a specific point. It’s useful in comparing things like costs, measurements, and trends.

Function Multiplication

The operation of function multiplication involves multiplying the outputs of two functions together. If \(f(x)\) and \(g(x)\) are your functions, then the product can be expressed as \((fg)(x)\). Evaluate both \(f(x)\) and \(g(x)\) first, and then multiply the results.

In our problem, \((fg)(-2)\) required us to find \(f(-2) = -4\) and \(g(-2) = 2\). Multiplying gives us -8.

To multiply functions:

• Identify x-value to consider.
• Determine the output for \(f(x)\) and \(g(x)\).
• Multiply these outputs to find the product.

Function multiplication can show how two independent processes interact multiplicatively, such as modeling joint probabilities or combined effects in physics or economics.

Function Division

Dividing two functions involves calculating the quotient of their outputs. When you have two functions, \(f(x)\) and \(g(x)\), their division is shown as \(\left(\frac{f}{g}\right)(x)\). Find each value first, then divide the outputs.

In our case, to evaluate \(\left(\frac{f}{g}\right)(0)\), we saw that \(f(0) = 8\) and \(g(0) = -1\). Dividing gives us -8.

Follow these steps for division:

• Select the x-value of interest.
• Calculate \(f(x)\) and \(g(x)\).
• Divide \(f(x)\) by \(g(x)\), ensuring \(g(x) eq 0\) to avoid undefined results.

Function division helps describe ratios or rates of change and is key in fields like economics for calculating efficiency or proportional relationships.