Exercise · ZW-IMAGE-AND-PREIMAGE-009 · ★★★ Advanced
Let \(f\colon\mathbb{R}\to\mathbb{R}\), \(f(x)=x^2-2x\). What is \(f^{-1}(\{-1\})\)?
Choose one option.
Show answer
- C. \(\{1\}\)
Explanation
Solve \(x^2-2x=-1\), which is equivalent to \((x-1)^2=0\), so \(x=1\) is the only solution and the preimage is \(\{1\}\). The other options fail on substitution: for example \(f(-1)=3\), not \(-1\).
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-008 What is f^-1(\y\) when y is not in f(A)?