Problem 4 - 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. \(\left(\frac{f}{g}\right)(x)\). $$ f(x)=x^{2}-2 ; g(x)=3 x $$

Step-by-Step Solution

Answer

1. \((f+g)(x)=x^2+3x-2\) 2. \((f-g)(x)=x^2-3x-2\) 3. \((f \cdot g)(x)=3x^3-6x\) 4. \(\left(\frac{f}{g}\right)(x)=\frac{x}{3}-\frac{2}{3x}, x \neq 0\)

Step 1: Adding Functions

To find \((f+g)(x)\), simply add the two functions: \(f(x) = x^2 - 2\) and \(g(x) = 3x\). This gives, \[(f+g)(x) = f(x) + g(x) = (x^2 - 2) + 3x = x^2 + 3x - 2.\]

Step 2: Subtracting Functions

To find \((f-g)(x)\), subtract \(g(x)\) from \(f(x)\): \[(f-g)(x) = f(x) - g(x) = (x^2 - 2) - 3x = x^2 - 3x - 2.\]

Step 3: Multiplying Functions

To find \((f\cdot g)(x)\), multiply the two functions: \[(f\cdot g)(x) = f(x) \cdot g(x) = (x^2 - 2)\cdot 3x.\]Distribute: \[(f\cdot g)(x) = 3x \cdot x^2 - 3x \cdot 2 = 3x^3 - 6x.\]

Step 4: Dividing Functions

To find \(\left(\frac{f}{g}\right)(x)\), divide \(f(x)\) by \(g(x)\): \[\left(\frac{f}{g}\right)(x) = \frac{x^2 - 2}{3x}.\] This can be simplified as \[ \frac{x^2}{3x} - \frac{2}{3x} = \frac{x}{3} - \frac{2}{3x}.\] Ensure in your final answer that \(x eq 0\) to avoid division by zero.

Key Concepts

Adding Functions

Adding functions involves combining two function expressions to form a new expression. Consider the functions \(f(x) = x^2 - 2\) and \(g(x) = 3x\). To add these functions, you calculate

• \((f+g)(x) = f(x) + g(x)\)
• Plug in the expressions: \((x^2 - 2) + 3x\)
• Simplify to get: \(x^2 + 3x - 2\)

This operation combines the terms of the functions, so make sure to keep track of like terms. The result is a new function, representing the sum of input values from the original functions. The new function, \(x^2 + 3x - 2\), provides a straightforward way to understand how two functions combine their effects on each value of \(x\).
Adding functions is particularly useful when you want to consider cumulative effects, like summing forces or costs modeled by separate functions.

Subtracting Functions

When subtracting functions, you focus on finding the difference between two function expressions. For \(f(x) = x^2 - 2\) and \(g(x) = 3x\), follow these steps:

• Calculate: \((f-g)(x) = f(x) - g(x)\)
• Insert the expressions: \((x^2 - 2) - 3x\)
• Simplify to obtain: \(x^2 - 3x - 2\)

When subtracting, keep an eye on the signs. It's crucial because subtracting affects both the terms and their coefficients. Subtraction reveals how one function counteracts or decreases the value of another across their domains. This skill is useful, for example, in scenarios where you're analyzing differences in elevation between two functions representing heights.

Multiplying Functions

Multiplying functions means determining how one function scales another. For \(f(x) = x^2 - 2\) and \(g(x) = 3x\), the process is as follows:

• Compute: \((f\cdot g)(x) = f(x) \cdot g(x)\)
• Apply the expressions: \((x^2 - 2) \cdot 3x\)
• Distribute to get: \(3x^3 - 6x\)

This operation is about combining effects—like how an increase in one factor scales another. The distributive property is your primary tool here, ensuring you apply each term in one expression to every term in the other.
Function multiplication is useful for modeling interactions between varying factors, like how changing weather might amplify seasonal effects on temperature, described by different functions.

Dividing Functions

Dividing functions involves determining how one function affects the scalings of another through division. Given \(f(x) = x^2 - 2\) and \(g(x) = 3x\), you handle this calculation by:

• Set up: \(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}\)
• Input the expressions: \(\frac{x^2 - 2}{3x}\)
• Simplify: \(\frac{x}{3} - \frac{2}{3x}\)

Ensure that the denominator is not zero to avoid undefined scenarios. Dividing functions helps understand ratios or rates, providing insight into how one variable proportionally affects another.
This operation is often seen in physics or economics, where one function might represent costs and the other quantities, enabling analysis of cost per unit or efficiency.