Problem 7 - Intermediate Algebra

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

\((f+g)(x) = 5x^2 - 3x\); \((f-g)(x) = -3x - 5x^2\); \((f \cdot g)(x) = -15x^3\); \(\left(\frac{f}{g}\right)(x) = -\frac{3}{5x}\) (for \(x \neq 0\)).

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

To find \((f+g)(x)\), we need to add the functions \(f(x)\) and \(g(x)\). Given, \(f(x) = -3x\) and \(g(x) = 5x^2\), the sum is: \((f+g)(x) = -3x + 5x^2\).

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

To find \((f-g)(x)\), we subtract \(g(x)\) from \(f(x)\). This gives us: \((f-g)(x) = -3x - 5x^2\).

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

The product \((f \cdot g)(x)\) involves multiplying \(f(x)\) by \(g(x)\). So, \((f \cdot g)(x) = (-3x) \cdot (5x^2) = -15x^3\).

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

To find the quotient \(\left(\frac{f}{g}\right)(x)\), divide \(f(x)\) by \(g(x)\): \(\left(\frac{f}{g}\right)(x) = \frac{-3x}{5x^2} = -\frac{3}{5x}\) when \(x eq 0\).

Key Concepts

Function Addition

When we talk about function addition, we are essentially combining two functions to form a new function. Given two functions, say \( f(x) \) and \( g(x) \), the addition of these functions is denoted as \((f+g)(x)\).
To find \((f+g)(x)\), simply add the expressions for \( f(x) = -3x \) and \( g(x) = 5x^2 \).

• Add corresponding parts of the functions
• Combine like terms if possible

This gives us the combined function as \(-3x + 5x^2\). Notice that when adding, you maintain the same domain for both functions, and the result holds for all \( x \) where both original functions are defined. Be aware of possible domain restrictions due to operations like division or square roots, though not applicable in this case.

Function Subtraction

Function subtraction deals with finding the difference between two functions. If you have functions \( f(x) \) and \( g(x) \), then \((f-g)(x)\) represents their difference. In other words, you subtract \( g(x) \) from \( f(x) \).
In our example, with \( f(x) = -3x \) and \( g(x) = 5x^2 \), the subtraction yields:

• Start with \( f(x) \)
• Subtract \( g(x) \)

This results in \(-3x - 5x^2\). Remember that in subtraction, just like in addition, the resulting function shares the domain of the original functions, unless specific operations dictate otherwise—such as dividing by a function that crosses zero.

Function Multiplication

Multiplying functions means creating a new function from the product of two given functions. When you see \((f \cdot g)(x)\), it tells you to multiply \( f(x) \) by \( g(x) \).
Consider \( f(x) = -3x \) and \( g(x) = 5x^2 \). To find their product, simply multiply:

• Directly multiply each term between the functions
• Simplify the expression

In this case, \((f \cdot g)(x) = (-3x)(5x^2) = -15x^3\). Multiplication of functions typically maintains the same domain as both original functions, with exceptions when the expressions involve undefined operations, such as division by zero.

Function Division

Function division is a bit unique because it involves dividing one function by another, represented as \( \left(\frac{f}{g}\right)(x) \). Here, it's crucial to note the conditions under which this division is permissible.
Given \( f(x) = -3x \) and \( g(x) = 5x^2 \), find their quotient by:

• Placing \( f(x) \) in the numerator
• Placing \( g(x) \) in the denominator
• Simplifying if possible

In this example, the result is \(-\frac{3}{5x}\) assuming \(x eq 0\) to avoid division by zero. Always identify the domain where the division is valid; here, it's all real \( x \) except where \( x = 0 \). Division functions may have restrictions due to zero denominators, keep this in mind when performing function division.