Problem 17 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find (a) \((f+g)(x),\) (b) \((f-g)(x)\) , (c) \((f g)(x),\) and \((d)(f / g)(x) .\) What is the domain of \(f / g ?\) $$f(x)=\frac{1}{x}, \quad g(x)=\frac{1}{x^{2}}$$

Step-by-Step Solution

Answer

The sum of the functions is \((f+g)(x) = \frac{x+1}{x^2}\). The difference of the functions is \((f-g)(x) = \frac{x-1}{x^2}\). The product of the functions is \((fg)(x) = \frac{1}{x^3}\). The quotient of the functions is \((f/g)(x) = x\). The domain of the function \((f/g)(x)\) is all real numbers except \(x = 0\).

Step 1: Add the Functions

To add two functions, simply add their equations. Given that \( f(x) = \frac{1}{x} \) and \( g(x) = \frac{1}{x^2} \), the addition of these two functions is: \((f+g)(x) = \frac{1}{x}+ \frac{1}{x^2} = \frac{x+1}{x^2}. \)

Step 2: Subtract the Functions

To subtract two functions, subtract their equations. Therefore, \((f-g)(x) = \frac{1}{x}- \frac{1}{x^2} = \frac{x-1}{x^2}. \)

Step 3: Multiply the Functions

To multiply two functions, multiply their equations. Therefore, \((fg)(x) = \frac{1}{x} \times \frac{1}{x^2} = \frac{1}{x^3}.\)

Step 4: Divide the Functions

To divide two functions, divide their equations. So, \((f/g)(x) = \frac{1}{x}/\frac{1}{x^2} = x. \)

Step 5: Find the Domain of the Division Function

The domain of the function \((f/g)(x)\) is the set of all real numbers, except for the values which make the denominator \(g(x)\) equal to zero. Since the denominator for \(g(x)\) is \( x^2 \), the function \(g(x)\) equals zero when \(x = 0\). Therefore, the domain of the resultant function \((f/g)(x)\) is all real numbers except for \(x = 0\).

Key Concepts

Function Addition

Function addition is like combining the results of two separate functions and finding a single output value. When you add functions, you're essentially adding together their equations or expressions.
For example, if you have two functions, such as \( f(x) = \frac{1}{x} \) and \( g(x) = \frac{1}{x^2} \), you perform function addition by calculating \((f+g)(x) = \frac{1}{x} + \frac{1}{x^2} \).
This will result in the simplified expression \( \frac{x+1}{x^2} \).
Function addition follows these basic steps:

• Write down the equations of the functions you want to add.
• Add them together, combining like terms if possible.
• Simplify the resulting expression.

Function Subtraction

Subtracting functions involves one function removing its values from another. This can be useful when you need to understand how one function differentiates from another.
Given the functions \( f(x) = \frac{1}{x} \) and \( g(x) = \frac{1}{x^2} \), the subtraction, represented as \((f-g)(x)\), is computed by
subtracting \( g(x) \) from \( f(x) \): \( \frac{1}{x} - \frac{1}{x^2} \).
This results in the expression \( \frac{x-1}{x^2} \).
Key points for function subtraction include:

• Always subtract the second function from the first as structured.
• Ensure that you simplify the resulting expression by combining like terms.

Function Multiplication

Function multiplication involves finding the product of two functions by multiplying their values together.
If we take \( f(x) = \frac{1}{x} \) and \( g(x) = \frac{1}{x^2} \), the multiplication yields \((fg)(x) = \frac{1}{x} \cdot \frac{1}{x^2} \).
Here, you multiply the numerators and denominators, leading to the result \( \frac{1}{x^3} \).
Multiplying functions is straightforward:

• Multiply the expressions of the functions directly.
• Simplify by combining the terms, similar to simplifying fractions.

Multiplication helps in scenarios where you want to find the combined rate of growth or other aggregate measures.

Function Division

When you divide functions, you are basically dividing the output of one function by another. This operation can show how one function behaves relative to another.
With functions \( f(x) = \frac{1}{x} \) and \( g(x) = \frac{1}{x^2} \), the division \((f/g)(x)\) corresponds to \( \frac{1/x}{1/x^2} \), simplifying to just \( x \).
Important steps in function division are:

• Divide the expression of the first function by that of the second function.
• Simplify to find the most reduced form of the expression.

Division of functions requires attention because you must consider the domain carefully, ensuring that you don't divide by zero.

Domain of a Function

Understanding the domain of a function is crucial because it tells you all the possible input values for which the function is defined.
Consider the division \((f/g)(x)\) with \( f(x) = \frac{1}{x} \) and \( g(x) = \frac{1}{x^2} \). The domain would exclude any values making \( g(x) \) equal zero, as this would result in division by zero.
Thus, for these functions, \( g(x) \) is zero at \( x = 0 \), implying the domain of \((f/g)(x)\) is all real numbers except \( x = 0 \).
To find the domain of a division function:

• Identify the values that make the denominator zero.
• Exclude those from the domain.

This understanding helps avoid undefined behaviors in mathematical operations, aiding accurate computations.