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Continuous vs Discrete Graph
publish date: 2026/05/23 18:30:37.767087 UTC
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If the domain of \(f(x) = 2x + 1\) is extended from \(\{0,1,2,3\}\) to all of \(\mathbb{R}\), how does the graph change?
Correct Answer
It becomes a continuous curve (a straight line) passing through all real-valued points
Explanation
Extending the domain to \(\mathbb{R}\) means every real number \(x\) has an image \(2x+1\), so the graph is the set \(\{(x, 2x+1) \mid x \in \mathbb{R}\}\) — an infinite collection of points lying on the straight line \(y = 2x + 1\). The four isolated points from the finite domain all lie on this line.
Reference
Introduction to Differential Calculus (Systematic Studies with Engineering Applications for Beginners) - 2012