Problem 6 - Algebra A Combined Function

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

Expressions are: 1. \(\sqrt[3]{x} + x - 3\), 2. \(\sqrt[3]{x} - x + 3\), 3. \(x^{1/3} \cdot (x-3)\), 4. \(\frac{\sqrt[3]{x}}{x-3}\), \(x \neq 3\).

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

To find \((f+g)(x)\), we need to add the expressions for \(f(x)\) and \(g(x)\). Given that \(f(x) = \sqrt[3]{x}\) and \(g(x) = x - 3\), we have: \[(f+g)(x) = f(x) + g(x) = \sqrt[3]{x} + (x-3) = \sqrt[3]{x} + x - 3.\]

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

To find \((f-g)(x)\), we subtract \(g(x)\) from \(f(x)\). Given \(f(x) = \sqrt[3]{x}\) and \(g(x) = x - 3\), perform the subtraction: \[(f-g)(x) = f(x) - g(x) = \sqrt[3]{x} - (x-3) = \sqrt[3]{x} - x + 3.\]

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

To find \((f \cdot g)(x)\), we multiply \(f(x)\) and \(g(x)\). With \(f(x) = \sqrt[3]{x}\) and \(g(x) = x - 3\), multiply the two functions: \[(f \cdot g)(x) = f(x) \cdot g(x) = \sqrt[3]{x} \cdot (x-3) = x^{1/3} \cdot (x-3).\]

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

To find \(\left(\frac{f}{g}\right)(x)\), divide \(f(x)\) by \(g(x)\). Using \(f(x) = \sqrt[3]{x}\) and \(g(x) = x - 3\), perform the division: \[\left(\frac{f}{g}\right)(x) = \frac{\sqrt[3]{x}}{x-3},\] with the restriction that \(x eq 3\) to avoid division by zero.

Key Concepts

Addition of Functions

When it comes to dealing with functions, one of the foundational operations you can perform is addition. This is as simple as adding two numbers together, but instead, you're combining entire functions.
Let's look at an example to clarify:

• Suppose you're working with two functions: \( f(x) = \sqrt[3]{x} \) and \( g(x) = x - 3 \).
• To add these functions, you simply take the function expressions and add them directly: \((f+g)(x) = f(x) + g(x)\).
• For these functions, it becomes: \( (f+g)(x) = \sqrt[3]{x} + (x-3) = \sqrt[3]{x} + x - 3 \).

This process is straightforward and gives you a new function that results from adding together all possible values of the original functions at each point of \( x \). Adding functions is particularly useful when you want to combine their behaviors into a single expression.

Subtraction of Functions

Subtraction of functions is similar to addition, but now you're taking one function and subtracting another. This operation is used when you wish to find how one function deviates from the other.
Here's how to subtract the functions step by step:

• With the same functions, \( f(x) = \sqrt[3]{x} \) and \( g(x) = x - 3 \), subtraction is performed as: \((f-g)(x) = f(x) - g(x)\).
• For our example, the operation looks like this: \((f-g)(x) = \sqrt[3]{x} - (x-3) = \sqrt[3]{x} - x + 3 \).

The outcome is a new function that reflects the difference between \( f(x) \) and \( g(x) \). This type of operation allows us to analyze contrasts and differences in various applications.

Multiplication of Functions

Multiplying functions involves multiplying the entire expressions of the functions. This is a bit more complex than simple arithmetic multiplication, as it combines the behaviors of the functions multiplicatively across all values of \( x \).
Here's how multiplication is performed:

• Using \( f(x) = \sqrt[3]{x} \) and \( g(x) = x - 3 \), the multiplication is done as follows: \((f \cdot g)(x) = f(x) \cdot g(x)\).
• This results in: \((f \cdot g)(x) = \sqrt[3]{x} \cdot (x-3) = x^{1/3} \cdot (x-3) \).

The resulting function describes how the output of one function scales the output of another. Multiplying functions is crucial in diverse fields such as physics and economics to model complex interactions systematically.

Division of Functions

When dividing functions, you take one function and see how it relates to another by division. This is typically done when you need to understand how many times one function's output "fits" into another's.
Dividing functions will involve the following steps:

• Choose your functions: \( f(x) = \sqrt[3]{x} \) and \( g(x) = x - 3 \).
• To divide them, apply: \( \left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} \).
• This results in: \( \left( \frac{f}{g} \right)(x) = \frac{\sqrt[3]{x}}{x-3} \).

A key cautious point: division by zero should be avoided, which means in our case, \( x eq 3 \) to prevent the denominator from becoming zero. Understanding division of functions is fundamental in calculus and real-world scenarios to explore comparative rates such as speed or growth.