Problem 2 - Algebra 2

Question

Find \((f+g)(x),(f-g)(x),(f \cdot g)(x),\) and \(\left(\frac{f}{g}\right)\) for each \(f(x)\) and \(g(x)\) $$ \begin{array}{l}{f(x)=x^{2}+3} \\ {g(x)=x-4}\end{array} $$

Step-by-Step Solution

Answer

1. \((f+g)(x) = x^2 + x - 1\) 2. \((f-g)(x) = x^2 - x + 7\) 3. \((f \cdot g)(x) = x^3 - 4x^2 + 3x - 12\) 4. \(\left( \frac{f}{g} \right)(x) = \frac{x^2 + 3}{x - 4}\)

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

The expression \( (f+g)(x) \) represents the sum of \( f(x) \) and \( g(x) \).\[(f+g)(x) = f(x) + g(x) = (x^2 + 3) + (x - 4)\]Combine the like terms for the result:\[(f+g)(x) = x^2 + x - 1\]

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

The expression \( (f-g)(x) \) represents the difference between \( f(x) \) and \( g(x) \).\[(f-g)(x) = f(x) - g(x) = (x^2 + 3) - (x - 4)\]Simplify by distributing the negative sign and combining like terms:\[(f-g)(x) = x^2 + 3 - x + 4 = x^2 - x + 7\]

Step 3: Calculate \( (f \, \cdot \, g)(x) \)

The expression \( (f \cdot g)(x) \) represents the product of \( f(x) \) and \( g(x) \).\[(f \cdot g)(x) = f(x) \cdot g(x) = (x^2 + 3)(x - 4)\]Apply the distributive property:\[(f \cdot g)(x) = x^2 \cdot x + x^2 \cdot (-4) + 3 \cdot x + 3 \cdot (-4)\]Simplify the expression:\[(f \cdot g)(x) = x^3 - 4x^2 + 3x - 12\]

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

The expression \( \left( \frac{f}{g} \right)(x) \) represents the quotient of \( f(x) \) and \( g(x) \).\[\left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} = \frac{x^2 + 3}{x - 4}\]This expression cannot be simplified further and is undefined for \( x = 4 \) since it makes the denominator zero.

Key Concepts

Function Operations

Function operations involve combining functions in various ways, such as addition, subtraction, multiplication, and division. These operations help us create new functions from existing ones and explore their properties. Understanding function operations allows us to manipulate functions to solve complex problems.

Here's a quick overview of the different types of function operations:

• Addition: Combining two functions by summing their output values for each input value.
• Subtraction: Finding the difference between the output values of two functions for each input.
• Multiplication: Obtaining a new function by multiplying the output values of two functions.
• Division: Forming a quotient function by dividing the output values of one function by another, with certain restrictions.

It's vital to ensure all operations are performed within the valid domain of the resulting function.

Addition of Functions

When adding two functions together, we're performing the operation of addition on their outputs. For functions \( f(x) \) and \( g(x) \), the sum is represented as \((f+g)(x) = f(x) + g(x)\). The process involves:

• Taking the output from each function for a given input \( x \).
• Summing these outputs to find the result for the new function.
• Simplifying where possible by combining like terms.

In the example, adding \((x^2 + 3)\) and \((x - 4)\) yielded \(x^2 + x - 1\), demonstrating how like terms (here, the constant and linear terms) simplify the expression. This result is valid for all \( x \).

Subtraction of Functions

Subtracting functions involves taking the output of one function and subtracting the output of another for the same input, creating a new function. Represented as \((f-g)(x) = f(x) - g(x)\), here's how it works:

• Consider the outputs of the given functions for any specific \( x \).
• Subtract the output of \( g(x) \) from \( f(x) \).
• Simplify the resulting expression by combining like terms.

In our example, subtracting \((x - 4)\) from \((x^2 + 3)\) resulted in \(x^2 - x + 7\). This operation requires careful handling of subtraction, especially with negative signs, to ensure correctness.

Multiplication of Functions

The multiplication of two functions involves multiplying their outputs for each input value. This operation is denoted as \((f \cdot g)(x) = f(x) \cdot g(x)\). Here's how to do it:

• Take each term from \( f(x) \), multiplying it by each term in \( g(x) \).
• Use the distributive property to expand the expression.
• Simplify by combining like terms to form a single polynomial function.

In our given example, multiplying \((x^2 + 3)\) by \((x - 4)\) resulted in \(x^3 - 4x^2 + 3x - 12\). The process requires methodical application of multiplication to ensure all terms are correctly considered.

Division of Functions

Dividing functions creates a quotient function where outputs of one function are divided by those of another, expressed as \(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}\). Follow these guidelines:

• Ensure the denominator \( g(x) \) is not zero for any \( x \) involved; division by zero is undefined.
• Write the expression as a fraction, simplifying where possible.
• Express any constraints where the division is undefined.

For our example with \( \frac{x^2 + 3}{x - 4} \), it remains unsimplified as a division of polynomials, and it's important to highlight that \( x eq 4 \), as it would make the denominator zero.