Exercise · ZW-IMAGE-AND-PREIMAGE-001 · ★☆☆ Basic
What is the image of \(X\subseteq A\) under \(f\colon A\to B\)?
Choose one option.
Show answer
- C. The set \(\{f(x)\colon x\in X\}\) of values \(f\) takes on \(X\)
Explanation
The image of \(X\) collects the values \(f(x)\) with \(x\in X\), and it is a subset of \(B\). The other options describe a zero set, a preimage, or the complement of the image, which are different objects.
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-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)?
- ZW-IMAGE-AND-PREIMAGE-009 Let fcolonℝ→ℝ, f(x)=x^2-2x. What is f^-1(\-1\)?