which statements correctly describe the graph of the function $f(x) = x^3 - 4x^2 - 3x + 18$? select three options.
- there are three unique solutions to the function when $y=0$.
- the function has a double root.
- as $x$ increases from negative infinity to positive infinity, the $y$-values increase, decrease, and then increase again.
- as $x$ approaches negative infinity, $y$ approaches positive infinity.
- the domain and range of the function are the set of real numbers.

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Accepted Answer

To solve this, we analyze each statement about the cubic function $ f(x) = x^3 - 4x^2 - 3x + 18 $: ### Analyzing Statement 1: "There are three unique solutions to the function when $ y = 0 $." A cubic function (degree 3) can have up to 3 real roots. Let's factor $ f(x) $: Try rational roots using Rational Root Theorem. Possible roots: $ \pm1, \pm2, \pm3, \pm6, \pm9, \pm18 $. - $ f(-2) = (-2)^3 - 4(-2)^2 - 3(-2) + 18 = -8 - 16 + 6 + 18 = 0 $. So $ (x + 2) $ is a factor. Divide $ f(x) $ by $ (x + 2) $: Using polynomial division or synthetic division: $$ \begin{array}{r|rrrr} -2 & 1 & -4 & -3 & 18 \ & & -2 & 12 & -18 \ \hline & 1 & -6 & 9 & 0 \ \end{array} $$ So $ f(x) = (x + 2)(x^2 - 6x + 9) = (x + 2)(x - 3)^2 $. The roots are $ x = -2 $ (simple root) and $ x = 3 $ (double root). So total real roots: 2 unique (but 3 roots with multiplicity). Wait, but the statement says "three unique solutions"—no, because $ x = 3 $ is a double root (not unique in value, but counts twice in roots). Wait, maybe misinterpretation: "solutions when $ y = 0 $" are the roots. The function has roots $ x = -2 $ and $ x = 3 $ (with multiplicity 2). So total roots: 3 (counting multiplicity), but unique roots: 2. Wait, maybe the statement is incorrect? Wait, no—wait, the graph of a cubic with a double root will cross the x-axis at the simple root and touch at the double root. So the number of x-intercepts (unique solutions) is 2, but the equation $ f(x) = 0 $ has 3 roots (with multiplicity). Wait, maybe the statement is "three solutions" (counting multiplicity) but it says "three unique solutions"—no. Wait, maybe I made a mistake. Wait, $ (x - 3)^2 = 0 $ has a double root, so the roots are $ x = -2, x = 3, x = 3 $. So unique solutions (distinct x-values) are 2, but total solutions (counting multiplicity) are 3. The statement says "three unique solutions"—that's incorrect? Wait, no—maybe the question means "three real solutions" (counting multiplicity). Wait, maybe the statement is intended to mean "three real solutions" (including multiplicity). If so, then it's correct. But let's check other statements. ### Analyzing Statement 2: "The function has a double root." From factoring, $ f(x) = (x + 2)(x - 3)^2 $, so $ x = 3 $ is a double root. So this statement is correct. ### Analyzing Statement 3: "As $ x $ increases from negative infinity to positive infinity, the $ y $-values increase, decrease, and then increase again." The derivative of $ f(x) $ is $ f'(x) = 3x^2 - 8x - 3 $. Let's find critical points: solve $ 3x^2 - 8x - 3 = 0 $. Discriminant: $ 64 + 36 = 100 $, so roots: $ x = \frac{8 \pm 10}{6} $, so $ x = \frac{18}{6} = 3 $ or $ x = \frac{-2}{6} = -\frac{1}{3} $. So the function's behavior: for $ x < -\frac{1}{3} $, $ f'(x) > 0 $ (function increasing); between $ -\frac{1}{3} $ and $ 3 $, $ f'(x) < 0 $ (function decreasing); for $ x > 3 $, $ f'(x) > 0 $ (function increasing). So as $ x $ goes from $ -\infty $ to $ \infty $, the function increases, then decreases, then increases. So this statement is correct. ### Analyzing Statement 4: "As $ x $ approaches negative infinity, $ y $ approaches positive infinity." For a cubic function $ ax^3 + bx^2 + cx + d $ with $ a > 0 $ (here $ a = 1 > 0 $), as $ x \to -\infty $, $ x^3 $ dominates, so $ y \to -\infty $. So this statement is incorrect. ### Analyzing Statement 5: "The domain and range of the function are the set of real numbers." A cubic function (polynomial) has domain $ \mathbb{R} $. For range: since it's a cubic with $ a > 0 $, as $ x \to \infty $, $ y \to \infty $; as $ x \to -\infty $, $ y \to -\infty $, so range is $ \mathbb{R} $. So this statement is correct. Wait, let's re-express: - Statement 1: "three unique solutions when $ y = 0 $"—the roots are $ x = -2 $ and $ x = 3 $ (double root). So unique solutions (distinct x-values) are 2, but total roots (counting multiplicity) are 3. If the statement means "three solutions (counting multiplicity)" then it's correct, but "unique" implies distinct. Maybe the statement is intended to say "three real solutions" (including multiplicity), so it's correct. - Statement 2: "double root"—correct, since $ (x - 3)^2 $. - Statement 3: "increases, decreases, increases"—correct, from derivative analysis. - Statement 4: "as $ x \to -\infty $, $ y \to +\infty $"—incorrect, since $ a = 1 > 0 $, so $ y \to -\infty $. - Statement 5: "domain and range are $ \mathbb{R} $"—correct, since cubic polynomials have domain $ \mathbb{R} $ and range $ \mathbb{R} $. Wait, but the question says "select three options". Let's check again: 1. "Three unique solutions when $ y = 0 $"—the roots are $ x = -2 $ (1), $ x = 3 $ (2, but double root). So unique solutions (distinct x-values) are 2, but total roots (counting multiplicity) are 3. So if "solutions" means roots (counting multiplicity), then it's 3 solutions (even though one is repeated). Maybe the statement is correct (interpreting "solutions" as roots, not unique x-values). 2. "Double root"—correct. 3. "Increases, decreases, increases"—correct. 4. "As $ x \to -\infty $, $ y \to +\infty $"—incorrect. 5. "Domain and range are $ \mathbb{R} $"—correct. Wait, now I'm confused. Let's re-express the function: $ f(x) = (x + 2)(x - 3)^2 $. - **Statement 1**: When $ y = 0 $, $ (x + 2)(x - 3)^2 = 0 $. Solutions: $ x = -2 $, $ x = 3 $, $ x = 3 $. So there are three solutions (counting multiplicity), but two unique x-values. If the statement says "three unique solutions", that's incorrect. But maybe the question has a typo, and "unique" is not intended. If we take "solutions" as roots (counting multiplicity), then it's three solutions, so the statement is correct. - **Statement 2**: "Double root"—correct, since $ (x - 3)^2 $ is a factor. - **Statement 3**: "As $ x $ increases from $ -\infty $ to $ +\infty $, $ y $-values increase, decrease, increase"—let's check the end behavior and critical points. The leading term is $ x^3 $, so as $ x \to +\infty $, $ y \to +\infty $; as $ x \to -\infty $, $ y \to -\infty $. The derivative $ f'(x) = 3x^2 - 8x - 3 $ has roots at $ x = -\frac{1}{3} $ and $ x = 3 $. So the function is increasing on $ (-\infty, -\frac{1}{3}) $, decreasing on $ (-\frac{1}{3}, 3) $, and increasing on $ (3, +\infty) $. So as $ x $ goes from $ -\infty $ to $ +\infty $, it increases (to $ x = -\frac{1}{3} $), then decreases (to $ x = 3 $), then increases (to $ +\infty $). So the statement is correct. - **Statement 4**: "As $ x $ approaches negative infinity, $ y $ approaches positive infinity"—incorrect, because leading term $ x^3 $, so $ y \to -\infty $. - **Statement 5**: "Domain and range of the function are the set of real numbers"—correct, because cubic polynomials are defined for all real $ x $ (domain $ \mathbb{R} $) and take all real values (range $ \mathbb{R} $). Now, we need to select three correct options. Let's check which three: - Statement 2: Correct. - Statement 3: Correct. - Statement 5: Correct. Wait, but what about Statement 1? If "solutions when $ y = 0 $" are the x-intercepts (unique), then there are two ( $ x = -2 $ and $ x = 3 $ ). But if "solutions" are the roots (counting multiplicity), then three. The statement says "three unique solutions"—"unique" implies distinct, so that's incorrect. So Statement 1 is incorrect. So correct statements: - Statement 2: "The function has a double root." (Correct, $ (x - 3)^2 $) - Statement 3: "As $ x $ increases from negative infinity to positive infinity, the $ y $-values increase, decrease, and then increase again." (Correct, from derivative analysis) - Statement 5: "The domain and range of the function are the set of real numbers." (Correct, cubic polynomials have domain $ \mathbb{R} $ and range $ \mathbb{R} $) Wait, but let's confirm Statement 5: For any cubic polynomial $ ax^3 + bx^2 + cx + d $ with $ a eq 0 $, the domain is $ \mathbb{R} $ (defined for all real $ x $) and the range is $ \mathbb{R} $ (since it's continuous and tends to $ \pm\infty $ as $ x \to \pm\infty $). So Statement 5 is correct. Statement 2: Correct (double root at $ x = 3 $). Statement 3: Correct (behavior from derivative: increasing, decreasing, increasing). So the three correct options are: - The function has a double root. - As $ x $ increases from negative infinity to positive infinity, the $ y $-values increase, decrease, and then increase again. - The domain and range of the function are the set of real numbers. Wait, but let's check Statement 1 again. If "solutions when $ y = 0 $" are the roots (counting multiplicity), then there are three solutions ( $ x = -2, 3, 3 $ ), so "three solutions" (not unique). Maybe the statement has a wording error, but if we take "solutions" as roots (counting multiplicity), then it's correct. But "unique" is misleading. Alternatively, maybe the intended correct options are: 1. The function has a double root. 2. As $ x $ increases from negative infinity to positive infinity, the $ y $-values increase, decrease, and then increase again. 3. The domain and range of the function are the set of real numbers. Or including Statement 1 if we ignore "unique". But let's verify with the graph: - A cubic with a double root (touching the x-axis at $ x = 3 $) and crossing at $ x = -2 $. - End behavior: as $ x \to +\infty $, $ y \to +\infty $; as $ x \to -\infty $, $ y \to -\infty $ (so Statement 4 is incorrect). - Domain: all real numbers (correct). - Range: all real numbers (correct, since cubic). - Double root: correct. - Behavior: increases, decreases, increases (correct). - Number of solutions when $ y = 0 $: three (counting multiplicity), so if "unique" is a mistake, then Statement 1 is correct. But "unique" means distinct, so it's incorrect. Given the options, the three correct ones are likely: - The function has a double root. - As $ x $ increases from negative infinity to positive infinity, the $ y $-values increase, decrease, and then increase again. - The domain and range of the function are the set of real numbers. Or including Statement 1 if "unique" is a misnomer. But based on strict interpretation, "unique solutions" would be distinct x-values, so Statement 1 is incorrect. So the three correct are 2, 3, 5. # Answer: - The function has a double root. - As $ x $ increases from negative infinity to positive infinity, the $ y $-values increase, decrease, and then increase again. - The domain and range of the function are the set of real numbers. (Or if Statement 1 is considered correct with "solutions" meaning roots counting multiplicity, then 1, 2, 3 or 1, 2, 5. But based on "unique", Statement 1 is incorrect. So the three correct are the ones with double root, the increasing-decreasing-increasing behavior, and domain/range as $ \mathbb{R} $.)

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Question

Explanation:

Analyzing Statement 1: "There are three unique solutions to the function when \( y = 0 \)."

Analyzing Statement 2: "The function has a double root."

Analyzing Statement 3: "As \( x \) increases from negative infinity to positive infinity, the \( y \)-values increase, decrease, and then increase again."

Analyzing Statement 4: "As \( x \) approaches negative infinity, \( y \) approaches positive infinity."

Analyzing Statement 5: "The domain and range of the function are the set of real numbers."

Answer: