Problem 1 - 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. \(\left(\frac{f}{g}\right)(x)\). $$ f(x)=x-7 ; g(x)=2 x+1 $$

Step-by-Step Solution

Answer

a. \(3x - 6\), b. \(-x - 8\), c. \(2x^2 - 13x - 7\), d. \(\frac{x - 7}{2x + 1}\).

Step 1: Define (f+g)(x)

The sum of the functions is defined as (f+g)(x)=f(x)+g(x). Substituting the given functions:\(f(x) = x - 7\) and \(g(x) = 2x + 1\).Perform the addition: \((f+g)(x) = (x - 7) + (2x + 1)\) which simplifies to \(3x - 6\).

Step 2: Define (f-g)(x)

The difference of the functions is given by \((f-g)(x) = f(x) - g(x)\). Substitute the functions:\(f(x) = x - 7\) and \(g(x) = 2x + 1\).Perform the subtraction: \((f-g)(x) = (x - 7) - (2x + 1)\). Simplifying this, we get \(-x - 8\).

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

The multiplication of the functions is \((f \cdot g)(x) = f(x) \cdot g(x)\). Substitute the functions:\(f(x) = x - 7\) and \(g(x) = 2x + 1\).Perform the multiplication:\((f \cdot g)(x) = (x - 7)(2x + 1)\).Expanding this, we get \(2x^2 + x - 14x - 7 = 2x^2 - 13x - 7\).

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

The division of the functions is \(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}\). Substitute the functions:\(f(x) = x - 7\) and \(g(x) = 2x + 1\).Perform the division:\(\left(\frac{f}{g}\right)(x) = \frac{x - 7}{2x + 1}\). This expression does not simplify further.

Key Concepts

Function Addition

In algebra, combining functions through addition is a straightforward process. It involves adding the outputs of the two functions for the same input value.
This is expressed as \( (f+g)(x) = f(x) + g(x) \). Let's break this down with our example functions:
\( f(x) = x - 7 \) and \( g(x) = 2x + 1 \).
To find \((f+g)(x)\), plug in the expressions for \(f(x)\) and \(g(x)\):

• Start with \((f+g)(x) = (x - 7) + (2x + 1)\).
• Combine like terms: \(x + 2x = 3x\) and \(-7 + 1 = -6\).
• Thus, \((f+g)(x) = 3x - 6\).

Function addition is as simple as adding their expressions and combining like terms.

Function Subtraction

Just like addition, subtraction of functions involves subtracting the outputs of one function from another for the same input value.
The notation for subtracting functions is \((f-g)(x) = f(x) - g(x)\). Let’s look at our example:
\( f(x) = x - 7 \) and \( g(x) = 2x + 1 \).Using these definitions:

• Write: \((f-g)(x) = (x - 7) - (2x + 1)\).
• Distribute the subtraction: \(- (2x + 1) = -2x - 1\).
• Combine terms: \(x - 2x = -x\) and \(-7 - 1 = -8\).
• Resulting function: \((f-g)(x) = -x - 8\).

Subtraction may require careful distribution of negative terms to ensure accurate calculations.

Function Multiplication

Multiplication of functions involves multiplying their outputs for the same input value. This is expressed mathematically as
\((f \cdot g)(x) = f(x) \cdot g(x)\). For our example functions, we have:
\( f(x) = x - 7 \) and \( g(x) = 2x + 1 \).Let's multiply the functions:

• Set up: \((f \cdot g)(x) = (x - 7)(2x + 1)\).
• Expand using distributive property:
• \(x \cdot 2x = 2x^2\).
• \(x \cdot 1 = x\).
• \(-7 \cdot 2x = -14x\).
• \(-7 \cdot 1 = -7\).
• Combine like terms: \(x - 14x = -13x\).
• Final result: \((f \cdot g)(x) = 2x^2 - 13x - 7\).

Function multiplication often leads to polynomial expressions that might need expansion and simplification.

Function Division

Dividing functions involves taking the ratio of their outputs for the same input value.
This is expressed as \(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}\). For our provided functions:
\( f(x) = x - 7 \) and \( g(x) = 2x + 1 \).To perform the division:

• Simply set up the division: \(\left(\frac{f}{g}\right)(x) = \frac{x - 7}{2x + 1}\).
• Ensure \(g(x)\), here \(2x + 1\), is not zero to avoid undefined expressions.

Function division results in a rational expression. Simplification isn't always possible, as shown here.