Problem 5 - Intermediate Algebra

Question

For the functions \(f\) and \(g,\) find \(a .(f+g)(x), b .(f-g)(x)\) \(c .(f \cdot g)(x),\) and \(d .\left(\frac{f}{g}\right)(x) .\) $$f(x)=\sqrt[3]{x}, g(x)=x+5$$

Step-by-Step Solution

Answer

a. \(\sqrt[3]{x} + x + 5\); b. \(\sqrt[3]{x} - x - 5\); c. \(x^{4/3} + 5x^{1/3}\); d. \(\frac{\sqrt[3]{x}}{x+5}\), \(x \neq -5\).

Step 1: Understanding the Functions

The given functions are \( f(x) = \sqrt[3]{x} \) and \( g(x) = x + 5 \). We need to perform operations between these functions as stated in the tasks: adding, subtracting, multiplying, and dividing.

Step 2: Adding the Functions

To find \((f+g)(x)\), we add the two functions: \( (f+g)(x) = \sqrt[3]{x} + (x + 5) = \sqrt[3]{x} + x + 5 \).

Step 3: Subtracting the Functions

To find \((f-g)(x)\), we subtract \( g(x) \) from \( f(x) \): \( (f-g)(x) = \sqrt[3]{x} - (x + 5) = \sqrt[3]{x} - x - 5 \).

Step 4: Multiplying the Functions

To find \((f \cdot g)(x)\), we multiply the two functions: \( (f \cdot g)(x) = \sqrt[3]{x} \cdot (x + 5) = \sqrt[3]{x} \cdot x + \sqrt[3]{x} \cdot 5 = x^{4/3} + 5x^{1/3} \).

Step 5: Dividing the Functions

To find \(\left(\frac{f}{g}\right)(x)\), we divide \( f(x) \) by \( g(x) \): \( \left(\frac{f}{g}\right)(x) = \frac{\sqrt[3]{x}}{x + 5} \). Ensure \( x eq -5 \) to avoid division by zero.

Key Concepts

Function Addition

Function addition involves combining two functions into one by adding them together. When we add two functions, we simply add their outputs for any given input value. For the functions given, which are

• \( f(x) = \sqrt[3]{x} \)
• \( g(x) = x + 5 \)

we look for

• \((f+g)(x) = f(x) + g(x)\).

So, the addition of these functions results in:\[ (f+g)(x) = \sqrt[3]{x} + x + 5 \]This expression shows that the output of the combined function is simply the cube root of \(x\), added to \(x\), and then 5 more units.

Function Subtraction

Function subtraction is when we subtract one function from another. This operation results in a new function defined as the difference between the two. For subtracted functions:

• \((f-g)(x) = f(x) - g(x)\).

Given the functions

• \( f(x) = \sqrt[3]{x} \)
• \( g(x) = x + 5 \)

the subtraction results in \[(f-g)(x) = \sqrt[3]{x} - x - 5\]
In this scenario, the expression means you take the cube root of \(x\), subtract \(x\), and then subtract 5. This operation is useful for finding the difference between outputs of two functions at any input value.

Function Multiplication

Function multiplication means multiplying the outputs of two functions for each input value. This multiplication results in

• \((f \cdot g)(x) = f(x) \times g(x)\).

Using our functions

• \( f(x) = \sqrt[3]{x} \)
• \( g(x) = x + 5 \)

the product of these functions is calculated as:\[(f \cdot g)(x) = \sqrt[3]{x} \cdot (x + 5) = x^{4/3} + 5x^{1/3}\]
Here, each term in the product is expanded, showing the different parts of the new function. It's as if we multiply each term of one expression by each term of the other and sum the results.

Function Division

Function division involves dividing the output of one function by the output of another function. The resulting function is

• \(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}\).

For the functions provided

• \( f(x) = \sqrt[3]{x} \)
• \( g(x) = x + 5 \)

the division between them is:\[ \left(\frac{f}{g}\right)(x) = \frac{\sqrt[3]{x}}{x + 5} \]
It's important to note that \(x\) should not be \(-5\) as that would cause division by zero, which is undefined. This expression represents how the output of \(f(x)\) relates to \(g(x)\) through division, giving the proportion or rate of change between the two functions.