Problem 23 - College Algebra

Question

For the following exercises, determine the end behavior of the functions. $$ f(x)=x^{2}\left(2 x^{3}-x+1\right) $$

Step-by-Step Solution

Answer

As \( x \to \,\infty \), \( f(x) \to \,\infty \); as \( x \to \,\-\infty \), \( f(x) \to \,\-\infty \).

Step 1: Identify the Leading Term

First, we need to find the leading term of the function as x approaches infinity or negative infinity. In the function \( f(x) = x^2(2x^3 - x + 1) \), distribute \( x^2 \) over the terms in the parentheses. The leading term comes from distributing \( x^2 \) to \( 2x^3 \), which results in \( 2x^5 \). Thus, the leading term is \( 2x^5 \).

Step 2: Determine the End Behavior for x Approaching Infinity

Now, look at the highest degree term, which is \( 2x^5 \). As \( x \to \,\infty \), \( 2x^5 \to \,\infty \). This means that the function \( f(x) \) will also approach infinity. Therefore, \( \,\lim_{{x \to \infty}} f(x) = \,\infty \).

Step 3: Determine the End Behavior for x Approaching Negative Infinity

Similarly, examine the behavior of the function as \( x \to \,\-fty \). Since the leading term is \( 2x^5 \) and the power of \( x \) is odd, as \( x \to \,\-fty \), \( 2x^5 \to \,\-fty \). Thus, \( f(x) \) will approach \( \,\-fty \) as well. Therefore, \( \,\lim_{{x \to -\infty}} f(x) = \,\-fty \).

Step 4: Summarize the End Behavior

Based on the previous steps, the function \( f(x) = x^2(2x^3 - x + 1) \) has the following end behavior: as \( x \to \,\infty \), \( f(x) \) approaches \( \,\infty \), and as \( x \to \,\-fty \), \( f(x) \) approaches \( \,\-fty \).

Key Concepts

Leading Term Identification

Identifying the leading term of a polynomial function is a key step in understanding its end behavior. In any polynomial, the leading term is the term with the highest power of the variable, which greatly influences how the function behaves as the variable moves towards positive or negative infinity. For instance, in the given function, \( f(x) = x^2(2x^3 - x + 1) \), we start by distributing \( x^2 \) across each of the terms inside the parenthesis. This operation is crucial because it reveals the highest power of \( x \) resulting from the expression.

• Distribute \( x^2 \) to each term in \((2x^3 - x + 1)\).
• The most significant combination is \( x^2 \times 2x^3 \), yielding \( 2x^5 \), which is the leading term.

In sum, the leading term, \( 2x^5 \), dictates the polynomial's end behavior, since as \( x \to \infty \) or \( x \to -\infty \), the lower power terms become insignificant.

Asymptotic Behavior

Asymptotic behavior is a description of how a function behaves as it approaches a specific value or infinity. For polynomial functions, the focus is often on how the function behaves as \( x \to \infty \) or \( x \to -\infty \).Since the leading term determines the end behavior of a polynomial, focusing on \( 2x^5 \), we can predict its path:

• As \( x \to \infty \), \( 2x^5 \to \infty \). This means that the polynomial function \( f(x) \) will also approach infinity.
• Conversely, as \( x \to -\infty \), the leading term \( 2x^5 \) decreases towards negative infinity because the exponent (5) is odd, so \( f(x) \to -\infty \).

Thus, the asymptotic behavior of \( f(x) = x^2(2x^3 - x + 1) \) is such that it increases without bound for large positive \( x \) and decreases without bound for large negative \( x \).

Polynomial Functions

Polynomial functions are expressions consisting of variables and coefficients, with operations involving only addition, subtraction, multiplication, and non-negative integer exponents of variables. Understanding their general structure helps in analyzing specific characteristics like end behavior:

• The degree of the polynomial, represented by the highest exponent, is pivotal in determining the shape of the graph and its end behavior.
• Coefficients of the leading term affect the graph's direction on a vertical axis - positive leading coefficients mean the function rises, while negative coefficients imply it falls as \( x \to \infty \).
• Odd vs. even degrees inform about symmetry; an odd-degree polynomial like \( f(x) = 2x^5 \) has different behavior at each end (one rises, the other falls).

Polynomials use a simple algebraic structure to demonstrate complex behavior, notably how they trend as \( x \) becomes very large positively or negatively. Mastery of these useful rules simplifies predictions about their graphs.