use the continuity of the absolute value function ($|x|$ is continuous for all values of $x$) to determine the interval(s) on which the following function is continuous.

\f(x) = |x^2 + 12x - 18|\

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the function is continuous on
(simplify your answer. type your answer in interval notation. use a comma to separate answers as needed.)

b. the function is not continuous anywhere.

Question

use the continuity of the absolute value function ($|x|$ is continuous for all values of $x$) to determine the interval(s) on which the following function is continuous.

\f(x) = |x^2 + 12x - 18|\

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the function is continuous on
(simplify your answer. type your answer in interval notation. use a comma to separate answers as needed.)

b. the function is not continuous anywhere.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Polynomial Continuity" ], "new_concepts": [ "Continuity of Composite Functions", "Absolute Value Continuity" ], "current_concepts": [ "Polynomial Continuity", "Continuity of Composite Functions", "Absolute Value Continuity" ] } ## Identify the inner and outer functions $$ \begin{aligned} &g(x) = x^2 + 12x - 18 \ &h(x) = |x| \ &f(x) = (h \circ g)(x) = |x^2 + 12x - 18| \end{aligned} $$ ## Determine continuity of the components $$ \begin{aligned} &g(x) \text{ is continuous on } (-\infty, \infty) \text{ (polynomial)} \ &h(x) \text{ is continuous on } (-\infty, \infty) \text{ (absolute value function)} \end{aligned} $$ ## Apply composite function continuity theorem $$ \begin{aligned} &f(x) = h(g(x)) \text{ is continuous on } (-\infty, \infty) \end{aligned} $$

A. The function is continuous on $(-\infty, \infty)$. (Simplify your answer. Type your answer in interval notation. Use a comma to separate answers as needed.)

B. The function is not continuous anywhere.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Calculus", "Continuity of Composite Functions" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the inner and outer functions

Determine continuity of the components

Apply composite function continuity theorem

Answer: