Exercise · ZW-IMAGE-AND-PREIMAGE-006 · ★★☆ Intermediate
With \(f(x)=x^2\) from \(\mathbb{R}\) to \(\mathbb{R}\), what is \(f^{-1}([1,4])\)?
Choose one option.
Show answer
- B. \([-2,-1]\cup[1,2]\)
Explanation
\(x^2\in[1,4]\) is equivalent to \(|x|\in[1,2]\), that is, \(x\in[-2,-1]\cup[1,2]\). The option \([-2,2]\) includes 0, whose square is outside \([1,4]\), and the option \([1,4]\) confuses the preimage with the set itself.
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-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)?
- ZW-IMAGE-AND-PREIMAGE-009 Let fcolonℝ→ℝ, f(x)=x^2-2x. What is f^-1(\-1\)?