Problem 8 - 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. \(\left(\frac{f}{g}\right)(x)\). $$ f(x)=4 x^{3} ; g(x)=-6 x $$

Step-by-Step Solution

Answer

a. \(4x^3 - 6x\), b. \(4x^3 + 6x\), c. \(-24x^4\), d. \(\frac{-2x^2}{3}\), \(x \neq 0\).

Step 1: Sum of Functions

To find \((f+g)(x)\), we need to add the functions \(f(x)\) and \(g(x)\) together. \(f(x) = 4x^3\) and \(g(x) = -6x\). So, \((f+g)(x) = f(x) + g(x) = 4x^3 + (-6x) = 4x^3 - 6x\).

Step 2: Difference of Functions

To find \((f-g)(x)\), subtract \(g(x)\) from \(f(x)\). \(f(x) = 4x^3\) and \(g(x) = -6x\). Thus, \((f-g)(x) = f(x) - g(x) = 4x^3 - (-6x) = 4x^3 + 6x\).

Step 3: Product of Functions

To find \((f\cdot g)(x)\), multiply \(f(x)\) by \(g(x)\). With \(f(x) = 4x^3\) and \(g(x) = -6x\), we have: \((f\cdot g)(x) = f(x) \cdot g(x) = 4x^3 \cdot (-6x) = -24x^4\).

Step 4: Quotient of Functions

For \(\left(\frac{f}{g}\right)(x)\), divide \(f(x)\) by \(g(x)\). Given \(f(x) = 4x^3\) and \(g(x) = -6x\), \(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{4x^3}{-6x} = \frac{-2x^2}{3}\), for \(x eq 0\).

Key Concepts

Function Operations

In algebra, functions are mathematical entities that help us model relationships between variables. You can perform operations with functions just like you do with numbers. This is called "function operations". Common operations include:

• Adding functions: combine outputs by summing them.
• Subtracting functions: find the difference in outputs.
• Multiplying functions: product of their outputs.
• Dividing functions: output of one divided by another (careful with division by zero!).

Function operations are useful in transforming and solving algebraic expressions. They give insights into how functions interact and change based on inputs. Practice these operations can enhance your skills in calculus and algebra.

Polynomial Functions

A polynomial function is an expression of more than one algebraic term, which includes variables (like \(x\)) raised to whole-number exponents. Our given functions, \(f(x) = 4x^3\) and \(g(x) = -6x\), are both polynomials:

• \(f(x)\) is a cubic polynomial because its highest exponent is 3.
• \(g(x)\) is a linear polynomial with an exponent of 1.

Polynomials are foundational in algebra and calculus due to their diverse properties and behaviors. They are easy to graph and solve and can vary greatly in shape and size. Understanding polynomial functions help in analyzing and predicting data trends.

Quotient of Functions

Dividing one function by another gives us the quotient of functions. The given expression is \(\left(\frac{f}{g}\right)(x)\), where \(f(x) = 4x^3\) and \(g(x) = -6x\). So:\[\left(\frac{f}{g}\right)(x) = \frac{4x^3}{-6x} = \frac{-2x^2}{3}\]The quotient is only valid when the denominator \(g(x)\) is not zero, which conforms to the rule that division by zero is undefined. In our example, make sure \(x eq 0\). This operation can be used to simplify complex algebraic expressions. It also comes in handy when working with rational functions for their analysis and graphing.

Product of Functions

When you multiply two functions, the result is the product of functions. In this case, we are multiplying \(f(x) = 4x^3\) by \(g(x) = -6x\):\[(f \cdot g)(x) = 4x^3 \times (-6x) = -24x^4\]This product of functions tells us how the outputs of two functions interact multiplicatively. The process involves multiplying coefficients and adding the powers of similar variables. Handling products of functions is useful in simplifying expressions and solving polynomials in algebra. It broadens the understanding of how changes in one function affect another.