Question

select the correct answer.

which pair of statements describes the end behavior of the graph of the function \\(f(x) = x^3 - 2x^2 - 5x + 6\\)?

• as \\(x\\) approaches negative infinity, \\(f(x)\\) approaches infinity. as \\(x\\) approaches infinity, \\(f(x)\\) approaches infinity.
• as \\(x\\) approaches negative infinity, \\(f(x)\\) approaches infinity. as \\(x\\) approaches infinity, \\(f(x)\\) approaches negative infinity.
• as \\(x\\) approaches negative infinity, \\(f(x)\\) approaches negative infinity. as \\(x\\) approaches infinity, \\(f(x)\\) approaches negative infinity.
• as \\(x\\) approaches negative infinity, \\(f(x)\\) approaches negative infinity. as \\(x\\) approaches infinity, \\(f(x)\\) approaches infinity.

Explanation:

Identify the leading term of the polynomial function

$$ f(x) = x^3 - 2x^2 - 5x - 6 $$

The leading term is \(x^3\), which has an odd degree (\(n = 3\)) and a positive leading coefficient (\(a_n = 1 > 0\)).

Determine the limit as \(x\) approaches negative infinity

$$ \lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} x^3 = -\infty $$

Determine the limit as \(x\) approaches positive infinity

$$ \lim_{x \to \infty} f(x) = \lim_{x \to \infty} x^3 = \infty $$

Answer:

• (A) As \(x\) approaches negative infinity, \(f(x)\) approaches infinity. As \(x\) approaches infinity, \(f(x)\) approaches infinity.
• (B) As \(x\) approaches negative infinity, \(f(x)\) approaches infinity. As \(x\) approaches infinity, \(f(x)\) approaches negative infinity.
• (C) As \(x\) approaches negative infinity, \(f(x)\) approaches negative infinity. As \(x\) approaches infinity, \(f(x)\) approaches negative infinity.
• (D) As \(x\) approaches negative infinity, \(f(x)\) approaches negative infinity. As \(x\) approaches infinity, \(f(x)\) approaches infinity. (Correct answer)