Exercise · ZW-IMAGE-AND-PREIMAGE-008 · ★☆☆ Basic
What is \(f^{-1}(\{y\})\) when \(y\) is not in \(f(A)\)?
Choose one option.
Show answer
- D. The empty set
Explanation
If no \(a\in A\) satisfies \(f(a)=y\), the preimage has no elements, so it is empty. This shows the preimage exists even when \(f\) has no inverse.
Review the concept: Image and preimage
Given \(f\colon A\to B\), the image of \(X\subseteq A\) is the set of values \(f(x)\) with \(x\in X\), and the preimage of \(Y\subseteq B\) is the set of \(a\in A\) whose value lies in \(Y\).
\[f(X)=\{f(x)\colon x\in X\},\qquad f^{-1}(Y)=\{a\in A\colon f(a)\in Y\}\]
More practice problems: image and preimage
- ZW-IMAGE-AND-PREIMAGE-001 What is the image of X⊆ A under fcolon A→ B?
- ZW-IMAGE-AND-PREIMAGE-002 Which subset of A is the preimage of Y⊆ B?
- ZW-IMAGE-AND-PREIMAGE-003 If fcolon A→ B is surjective, what is f(A)?
- ZW-IMAGE-AND-PREIMAGE-004 For Y⊆ B, what is always true of f(f^-1(Y))?
- ZW-IMAGE-AND-PREIMAGE-005 Let f(x)=x^2 from ℝ to ℝ. What is f([-1,2])?
- ZW-IMAGE-AND-PREIMAGE-006 With f(x)=x^2 from ℝ to ℝ, what is f^-1([1,4])?
- ZW-IMAGE-AND-PREIMAGE-007 Let fcolon A→ B and X_1,X_2⊆ A. Which statement about images is correct?
- ZW-IMAGE-AND-PREIMAGE-009 Let fcolonℝ→ℝ, f(x)=x^2-2x. What is f^-1(\-1\)?