Problem 79 - A Graphical Approach to College Algebra

Question

Let \(f(x)=x^{3} .\) Evaluate each expression. $$f(2)$$

Step-by-Step Solution

Answer

\(f(2) = 8\).

Step 1: Understand the Function

Begin by understanding the function given, which is \(f(x) = x^3\). This function represents a cubic polynomial of \(x\), where the output is the cube of the input \(x\).

Step 2: Substitute the Value into the Function

Substitute the given value, 2, into the function \(f(x)\). This involves replacing \(x\) in the expression \(x^3\) with the value 2, giving us \(f(2) = 2^3\).

Step 3: Calculate the Cube

Calculate the cube of 2. This means multiplying 2 by itself three times: \(2 \times 2 \times 2 = 8\).

Step 4: Finalize the Answer

Having calculated \(2^3\), the value of \(f(2)\) is determined. Therefore, \(f(2) = 8\).

Key Concepts

Polynomial Functions

Polynomial functions are a fundamental concept in mathematics. They are equations that involve only non-negative integer power terms. A polynomial of degree \( n \) has a general form given by \( a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0 \), where \( a_n, a_{n-1}, \cdots, a_0 \) are constants and \( n \) is a non-negative integer.
Cubic functions, like the one in our exercise, are polynomials of degree 3, meaning the highest power of \( x \) is 3.
In the function \( f(x) = x^3 \), the leading term is \( x^3 \). This indicates that the relationship is proportional to the cube of \( x \).
Understanding polynomial functions is crucial because they are used extensively in both calculus and algebra to describe various phenomena.

Function Evaluation

Evaluating a function means finding the value of the function for a specific input. In mathematical terms, this involves substituting the input value into the function's formula to calculate the corresponding output.
To evaluate the cubic function \( f(x) = x^3 \) for \( f(2) \), the process involves:

• Recognizing the function form \( f(x) \)
• Substituting the given value \( x = 2 \)
• Calculating the result of the expression \( 2^3 \)

This procedure ensures that you're accurately following the mathematical operations needed to find the solution. Additionally, function evaluation helps in understanding how input values affect the output of a function.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The base is the number that is being multiplied, while the exponent indicates how many times the base is multiplied by itself.
In our problem, exponentiation comes into play when evaluating \( 2^3 \). Here, 2 is the base, and 3 is the exponent, which means we multiply 2 by itself three times: \( 2 \times 2 \times 2 = 8 \).
This operation is essential for dealing with polynomial functions and is a foundational skill in algebra. Exponents allow us to express repeated multiplication in a simplified form and are encountered frequently across various mathematical applications, from simple algebra to more complex calculus problems.