Problem 68 - Contemporary Precalculus

Question

The rule of a function \(f\) is given. Write an algebraic formula for \(f(x)\). Cube the input, add \(6,\) and divide the result by 5

Step-by-Step Solution

Answer

Answer: The algebraic formula for the function \(f(x)\) is \(f(x) = \frac{x^3 + 6}{5}\).

Step 1: Cube the input

First, we need to cube the input value, which is \(x\). The cube of a number can be written as that number raised to the power of \(3\). So, cubing the input \(x\) would be \(x^3\).

Step 2: Add 6 to the result

After cubing the input, we need to add \(6\). So, taking the result from Step 1 which is \(x^3\), we add 6 to it: \(x^3 + 6\).

Step 3: Divide the result by 5

Finally, we need to divide the result from Step 2 by 5. So, we take the result \((x^3 + 6)\) and divide by 5. Therefore, the algebraic formula for the function \(f(x)\) is: \(f(x) = \frac{x^3 + 6}{5}\)

Key Concepts

Function Notation

Function notation is a way of representing relationships between inputs and outputs in algebra through a function. It's a precise and efficient way to communicate calculations.

• The function notation often uses the letter "f" to denote a function, although other letters can be used.
• The notation "f(x)" reads as "f of x" and represents the value of the function at the input "x".

When you write a function like "f(x)", you're telling someone: "take x, do something to it, and this is the result." In our case, the 'something' is cubing "x", adding 6, and dividing by 5 to get the final result.

Cubic Functions

Cubic functions are polynomial functions of degree three. They are written in the form of:\[ f(x) = ax^3 + bx^2 + cx + d \]This specific type of function is essential in algebra as it reveals how rapid changes can occur in outputs compared to linear or quadratic functions.

• In our example, the expression "x^3" indicates the cubic relationship between "x" and "f(x)".
• Cubic functions can change direction up to two times, creating a curve on a graph.
• They are smooth and continuous, meaning they have no breaks or holes in their graph.

Cubic functions are especially interesting because they often model real-world phenomena where effects increase significantly as inputs change.

Mathematical Operations

In creating complex expressions and functions, mathematical operations such as addition, multiplication, and division are combined in structured ways.

• First, cubing the input involves using an exponent or power. Here, "x" is multiplied by itself twice (\( x \times x \times x \)).
• Second, adding a constant like "6" shifts the cubic graph upwards by the same amount.
• Finally, dividing the entire expression by "5" scales it down, affecting how steeply the graph of the function rises and falls.

Each step of the arithmetic operations is fundamental. These operations work together to build complex expressions like our function \( f(x) = \frac{x^3 + 6}{5} \). Each part of the operation modifies the function's characteristics in its unique way.