Restrict the Domain of the Function F So That the Function

question 3

Multiple Choice

Restrict the domain of the function f so that the function is one-to-one and has an in- verse function. Then find the inverse function $f ^ { - 1 }$ . State the domains and ranges of f and $f ^ { - 1 }$ . $f ( x ) = ( x - 2 ) ^ { 2 }$

• A) \(f ^ { - 1 } ( x ) = \sqrt { x } - 2\)
• The domain of \(f\) and the range of \(f ^ { - 1 }\) are all real numbers \(x\) such that \(x \geq 2\).
• The domain of \(f ^ { - 1 }\) and the range of \(f\) are all real numbers \(x\) such that \(x \geq 0\).
• B) \(f ^ { - 1 } ( x ) = \sqrt { x } + 2\)
• The domain of \(f\) and the range of \(f ^ { - 1 }\) are all real numbers \(x\) such that \(x \geq 0\).
• The domain of \(f ^ { - 1 }\) and the range of \(f\) are all real numbers \(x\) such that \(x \geq - 2\).
• C) \(f ^ { - 1 } ( x ) = \sqrt { x } + 2\)
• The domain of \(f\) and the range of \(f ^ { - 1 }\) are all real numbers \(x\) such that \(x \geq 2\).
• The domain of \(f ^ { - 1 }\) and the range of \(f\) are all real numbers \(x\) such that \(x \geq 0\).
• D) \(f ^ { - 1 } ( x ) = \sqrt { x } + 2\)
• The domain of \(f\) and the range of \(f ^ { - 1 }\) are all real numbers \(x\) such that \(x \geq 0\)
• The domain of \(f ^ { - 1 }\) and the range of \(f\) are all real numbers \(x\) such that \(x \geq 2\).
• E) \(f ^ { - 1 } ( x ) = \sqrt { x } - 2\)
• The domain of \(f\) and the range of \(f ^ { - 1 }\) are all real numbers \(x\) such that \(x \geq - 2\).
• The domain of \(f ^ { - 1 }\) and the range of \(f\) are all real numbers \(x\) such that \(x \geq 0\).

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Definitions:

Deductive Arguments

A form of reasoning where the conclusion is logically derived from premises that are assumed to be true.

Inductive Arguments

Reasoning that moves from specific instances to a general conclusion, often used to infer probabilities.

Conditional Statements

Logical statements that have an "if-then" structure, indicating that if one condition is true, then a specific result follows.

Claims

Statements or assertions that something is the case, typically without providing evidence or proof.