if the function $f(x) = mx + b$ has an inverse function, which statement must be true?

$m \
eq 0$
$m = 0$
$b \
eq 0$
$b = 0$

Question

if the function $f(x) = mx + b$ has an inverse function, which statement must be true?

$m \
eq 0$
$m = 0$
$b \
eq 0$
$b = 0$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Inverse Functions", "One-to-One Functions", "Linear Functions" ], "new_concepts": [], "current_concepts": [ "Inverse Functions", "One-to-One Functions", "Linear Functions" ] } ## Analyze the condition for a linear function to have an inverse $$ \begin{aligned} &f(x) = mx + b \text{ has an inverse if and only if it is a one-to-one function.} \end{aligned} $$ ## Evaluate the effect of the slope $m$ on the one-to-one property $$ \begin{aligned} &\text{If } m = 0, \text{ then } f(x) = b \text{ (a constant function), which is not one-to-one.}\ &\text{If } m eq 0, \text{ then } f(x) \text{ is strictly increasing or strictly decreasing, which is one-to-one.} \end{aligned} $$ ## Determine the required condition $$ \begin{aligned} &m eq 0 \end{aligned} $$

(A) $m eq 0$

(B) $m = 0$

(C) $b eq 0$

(D) $b = 0$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Inverse Functions" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze the condition for a linear function to have an inverse

Evaluate the effect of the slope \(m\) on the one-to-one property

Determine the required condition

Answer: