Problem 6 - Intermediate Algebra

Question

For the functions \(f\) and \(g,\) find \(a .(f+g)(x), b .(f-g)(x)\) \(c .(f \cdot g)(x),\) and \(d .\left(\frac{f}{g}\right)(x) .\) $$f(x)=\sqrt[3]{x}, g(x)=x-3$$

Step-by-Step Solution

Answer

\((f+g)(x) = \sqrt[3]{x} + x - 3\), \((f-g)(x) = \sqrt[3]{x} - x + 3\), \((f \cdot g)(x) = x^{\frac{4}{3}} - 3x^{\frac{1}{3}}\), \(\left(\frac{f}{g}\right)(x) = \frac{\sqrt[3]{x}}{x - 3}\).

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

The function \((f+g)(x)\) represents the sum of the functions \(f(x)\) and \(g(x)\). Calculate it by adding \(f(x)\) and \(g(x)\): \((f+g)(x) = f(x) + g(x) = \sqrt[3]{x} + (x - 3) \). Simplifying gives \((f+g)(x) = \sqrt[3]{x} + x - 3\).

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

The function \((f-g)(x)\) represents the difference between the functions \(f(x)\) and \(g(x)\). Calculate it by subtracting \(g(x)\) from \(f(x)\): \((f-g)(x) = f(x) - g(x) = \sqrt[3]{x} - (x - 3) \). Simplifying gives \((f-g)(x) = \sqrt[3]{x} - x + 3\).

Step 3: Evaluate \((f \cdot g)(x)\)

The function \((f \cdot g)(x)\) represents the product of the functions \(f(x)\) and \(g(x)\). Calculate it by multiplying \(f(x)\) and \(g(x)\): \((f \cdot g)(x) = f(x) \cdot g(x) = \sqrt[3]{x} \cdot (x - 3) \). This gives \((f \cdot g)(x) = x^{\frac{4}{3}} - 3x^{\frac{1}{3}}\).

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

The function \(\left(\frac{f}{g}\right)(x)\) represents the quotient of the functions \(f(x)\) and \(g(x)\). Calculate it by dividing \(f(x)\) by \(g(x)\): \(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{\sqrt[3]{x}}{x - 3}\). The expression remains in this form as it cannot be simplified further.

Key Concepts

Function Addition

In mathematics, function addition involves combining two functions into a single function by adding their corresponding outputs. This is similar to adding numbers, but applied to functions. Let's break it down with an example using the functions provided, \( f(x) = \sqrt[3]{x} \) and \( g(x) = x - 3 \).
The function \((f+g)(x)\) represents the sum of \( f(x) \) and \( g(x) \). You compute it by:

• Evaluating \( f(x) + g(x) = \sqrt[3]{x} + (x - 3) \)
• This simplifies to \( \sqrt[3]{x} + x - 3 \)

Function addition allows you to combine characteristics of both functions into one, enabling better analysis or predictions with their combined effects. Keep in mind that for \( (f+g)(x) \) to be defined everywhere, both functions need to have outputs for that specific \( x \).

Function Subtraction

Function subtraction is similar to function addition, but instead of adding the outputs of two functions, you subtract them. This can be useful when you need to find the difference between two mathematical processes or phenomena. Consider \( f(x) = \sqrt[3]{x} \) and \( g(x) = x - 3 \).
To obtain \((f-g)(x)\), you perform the subtraction:

• Calculate \( f(x) - g(x) = \sqrt[3]{x} - (x - 3) \)
• This simplifies to \( \sqrt[3]{x} - x + 3 \)

Subtraction can highlight differences between functions. It is particularly useful in comparing rates of change or deviations between two datasets. Always remember to consider the domains of the functions involved to ensure you are subtracting a defined output for the given \( x \) values.

Function Multiplication

Function multiplication involves multiplying the outputs of two functions for a given input \( x \). This can reveal interactions between factors represented by the functions. Let’s look at \( f(x) = \sqrt[3]{x} \) and \( g(x) = x - 3 \) to illustrate this concept. By performing function multiplication, you get \((f \cdot g)(x)\):

• Calculate \( f(x) \cdot g(x) = \sqrt[3]{x} \cdot (x - 3) \)
• Resulting in \( x^{\frac{4}{3}} - 3x^{\frac{1}{3}} \)

Multiplying functions gives insight into how one variable might influence another multiplicatively. When performing this operation, check that multiplication is defined at the input values you are interested in to avoid errors, like undefined points.

Function Division

Function division is the process of dividing the outputs of two functions. This is used to compare rates or find how one variable scales relative to another. Taking \( f(x) = \sqrt[3]{x} \) and \( g(x) = x - 3 \), division is shown by:

• Evaluate \( \left(\frac{f}{g}\right)(x) = \frac{\sqrt[3]{x}}{x - 3} \)

Unlike addition or multiplication, division can encounter undefined values more often. For example, \((f/g)(x)\) is undefined when \( g(x) = 0 \), which happens here when \( x = 3 \).
Thus, it's crucial to consider the domain and potential asymptotes or discontinuities this operation might introduce. Overall, function division emphasizes how one aspect of a problem might be diminished or augmented by another, providing valuable insights into proportional relationships.