91Ó°ÊÓ

When we talk about the domain of a polynomial function, we focus on all the possible values that we can use for the input, often represented by the variable \( x \). For polynomial functions, this is quite straightforward. The domain of a polynomial function is always the set of all real numbers. This means you can plug in any real number value for \( x \).

• There are no fractions to worry about causing division by zero.
• There are no square roots that could result in an imaginary number.

Simply put, polynomial functions are friendly in terms of what numbers you can input, allowing for an unlimited range of \( x \)-values without restriction. That's why their domain is represented as \( (-\infty, \infty) \). In this exercise, for the function \( f(x) = 5x^3 + 6x^2 - 1 \), its domain is just that: all real numbers.

The range of a function deals with the outputs, or \( y \)-values, that come from plugging every possible \( x \)-value from the domain into the function. For polynomial functions, the range can be a bit trickier to determine compared to the domain.

• For linear and cubic polynomials, like in the exercise, the range is the set of all real numbers \( (-\infty, \infty) \).
• This is because these functions continue infinitely in both the upward and downward directions.

For the specific function \( f(x) = 5x^3 + 6x^2 - 1 \), since it is cubic and has a positive leading coefficient, the graph rises to infinity as \( x \) moves to the right and falls to negative infinity as \( x \) moves to the left, covering every possible \( y \)-value. Thus, its range is also all real numbers.

Cubic functions are an exciting class of polynomial functions where the highest power of the variable \( x \) is three. These functions generally take the form \( ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are constants, and \( a eq 0 \).

• The graph of a cubic function typically forms an 'S' shape, clearly visible when plotted.
• This distinctive shape reflects the fact that a cubic function can have one real root or three real roots, and it can change direction at most twice.

In the given exercise, the function \( f(x) = 5x^3 + 6x^2 - 1 \) is a cubic function. The term \( 5x^3 \) controls the overall growth and end behavior of the graph. It dictates that as \( x \) heads towards positive infinity, \( f(x) \) rises without bound. Similarly, as \( x \) moves towards negative infinity, \( f(x) \) falls endlessly downward. This is why both the domain and range for cubic functions tend to be all real numbers.