worksheet 8: 3.5 & 4.1

1). determine whether the function $f(x) = x^4 - 5$ is one-to-one.

Question

worksheet 8: 3.5 & 4.1

1). determine whether the function $f(x) = x^4 - 5$ is one-to-one.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "One-to-One Functions", "Horizontal Line Test" ], "current_concepts": [ "One-to-One Functions", "Horizontal Line Test" ] } ## Define the algebraic condition for a one-to-one function $$ f(a) = f(b) \implies a = b $$ ## Test the function with counterexample values $$ \begin{aligned} f(1) &= (1)^4 - 5 = -4 \ f(-1) &= (-1)^4 - 5 = -4 \end{aligned} $$ ## Evaluate the one-to-one condition $$ f(1) = f(-1) \text{ but } 1 eq -1 $$ The function $f(x) = x^4 - 5$ is not one-to-one. { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "One-to-One Functions" ] }

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Explanation:

Define the algebraic condition for a one-to-one function

Test the function with counterexample values

Evaluate the one-to-one condition

Answer: