Problem 7 - 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. \(\left(\frac{f}{g}\right)(x)\). $$ f(x)=-3 x ; g(x)=5 x^{2} $$

Step-by-Step Solution

Answer

a. \((f+g)(x) = 5x^2 - 3x\); b. \((f-g)(x) = -5x^2 - 3x\); c. \((f \cdot g)(x) = -15x^3\); d. \(\left(\frac{f}{g}\right)(x) = \frac{-3}{5x}\).

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

To find \((f+g)(x)\), add the expressions for \(f(x)\) and \(g(x)\). Here, \(f(x) = -3x\) and \(g(x) = 5x^2\). So, \((f+g)(x) = f(x) + g(x) = -3x + 5x^2\). Write this in a standard polynomial form: \((f+g)(x) = 5x^2 - 3x\).

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

To find \((f-g)(x)\), subtract \(g(x)\) from \(f(x)\). So, \((f-g)(x) = f(x) - g(x) = -3x - 5x^2\). Rearranging, we have \((f-g)(x) = -5x^2 - 3x\).

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

To find \((f \cdot g)(x)\), multiply \(f(x)\) by \(g(x)\). So, \((f \cdot g)(x) = (-3x)(5x^2) = -15x^3\).

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

To find \(\left(\frac{f}{g}\right)(x)\), divide \(f(x)\) by \(g(x)\). So, \(\left(\frac{f}{g}\right)(x) = \frac{-3x}{5x^2}\). Simplifying, divide the numerator and the denominator by \(x\), which gives \(\left(\frac{f}{g}\right)(x) = \frac{-3}{5x}\).

Key Concepts

Function Addition

Function addition is simply the process of combining two functions by adding their respective outputs. In our exercise, we have the functions:

• \( f(x) = -3x \)
• \( g(x) = 5x^2 \)

To find \((f+g)(x)\), we add these two functions:\[(f+g)(x) = f(x) + g(x) = -3x + 5x^2\].
It's important to write the resulting function in standard polynomial form, which prioritizes descending powers of \(x\). So, \((f+g)(x) = 5x^2 - 3x\).
This combination reveals a new polynomial function showcasing the traits of both \(f\) and \(g\). Function addition helps understand how individual functions contribute to a resultant function.

Function Subtraction

Function subtraction involves subtracting one function's output from another function's output. In this exercise, we subtract the function \(g(x)\) from \(f(x)\):

• \( (f-g)(x) = f(x) - g(x) = -3x - 5x^2 \)

Rearrange terms to maintain the standard polynomial form, resulting in: \((f-g)(x) = -5x^2 - 3x\).
Function subtraction can expose differences between two functions and how one function might offset or enhance certain values of the other.

Function Multiplication

To understand function multiplication, it's key to realize that multiplying two functions' formulas combines their effects multiplicatively. Here, we multiply \(f(x)\) and \(g(x)\):

• \( (f \cdot g)(x) = (-3x)(5x^2) = -15x^3 \)

The product is a new function that incorporates both variables' effects from \(f(x)\) and \(g(x)\), raising the degree of the polynomial.
This further answers how the rate of change of one function affects the other, producing a more complex result. Function multiplication yields insight into how multiplicative interactions between functions transform their outputs.

Function Division

Function division involves dividing the output of one function by another. For the functions given, we need to divide \(f(x)\) by \(g(x)\):

• \( \left(\frac{f}{g}\right)(x) = \frac{-3x}{5x^2} \)

Simplifying involves canceling common terms. Here, we divide numerator and denominator by \(x\), resulting in: \( \left(\frac{f}{g}\right)(x) = \frac{-3}{5x} \).
Function division is beneficial in understanding the relative behaviors of two functions, especially revealing the scaling and inversely proportional aspects influenced by division.
Considerations include ensuring \(g(x)\) doesn't equal zero, as this would make the division undefined.