91Ó°ÊÓ

A cubic function is a type of polynomial function that has a degree of three. This means that the highest power of the variable, usually denoted as x, is three. The general form of a cubic function is given by \( f(x) = ax^3 + bx^2 + cx + d \), where a, b, c, and d are constants and the coefficient a is not zero.

Cubic functions are interesting because they can have up to three real roots and their graphs have a distinctive S-shape, which may include one or two turning points. In our exercise, the cubic function we are working with is \( f(x) = x^3 \). This is a simpler example where a, b, c, and d are set such that only the term \( x^3 \) is present, making it easy to evaluate at any point.

Function evaluation involves finding the value of the function at a specific input value. In other words, it's about inserting a number into the function and computing the result. This is commonly represented as \( f(a) \) where a is the input value.

For our exercise, we are asked to evaluate the function \( f(x) = x^3 \) at \( x = 4 \). To do this, we replace every x in the function with 4. Therefore, it becomes \( f(4) = 4^3 \).
If the function were more complex with multiple terms, you would perform a similar substitution for each term and then combine the results according to the function's operations.

Substitution is a method used in mathematics where one replaces a variable with a specific value to simplify or evaluate equations and functions. In our exercise, substitution is used to evaluate the function \( f(x) = x^3 \) at \( x = 4 \).

The steps are straightforward:

• Identify the specific value to substitute. Here, it is 4.
• Replace the variable x in the function with this value. Hence, \( f(4) = 4^3 \).
• Calculate the result. In this case, calculating \( 4^3 \) involves performing the multiplication: \( 4 \times 4 \times 4 = 64 \).

This step-by-step process ensures that we correctly evaluate the function at any given point.