Let m and b be real numbers, and consider the following three functions: f(x) = 2x + 1, g(x) = − 3x + 2, and h(x) = mx + b. A. If function f has a codomain of (5, 7) U (7, 9), the largest its domain can be chosen is (2, 3) U (3, 4). Explain. B. If function g has a codomain of (-1, 1) U (1, 3), the largest its domain can be chosen is (-3, 3) U (3, 9). Explain. C. Within the context of the e- definition of a limit, your result from part A suggests that if € is equal to 2, the largest that can be chosen for function f(x) is 1. Explain. D. Within the context of the e-8 definition of a limit, your result from part B suggests that if € is equal to 2, the largest that can be chosen for function g(x) is 6. Explain. E. The following statement is (ever so slightly) false: "Within the context of the e-8 definition of a limit, the results from parts B and C suggest that for a given positive real number €, the largest that can be chosen for function h(x) is" Modify this statement (ever so slightly) so that it is true, and
Let m and b be real numbers, and consider the following three functions: f(x) = 2x + 1, g(x) = − 3x + 2, and h(x) = mx + b. A. If function f has a codomain of (5, 7) U (7, 9), the largest its domain can be chosen is (2, 3) U (3, 4). Explain. B. If function g has a codomain of (-1, 1) U (1, 3), the largest its domain can be chosen is (-3, 3) U (3, 9). Explain. C. Within the context of the e- definition of a limit, your result from part A suggests that if € is equal to 2, the largest that can be chosen for function f(x) is 1. Explain. D. Within the context of the e-8 definition of a limit, your result from part B suggests that if € is equal to 2, the largest that can be chosen for function g(x) is 6. Explain. E. The following statement is (ever so slightly) false: "Within the context of the e-8 definition of a limit, the results from parts B and C suggest that for a given positive real number €, the largest that can be chosen for function h(x) is" Modify this statement (ever so slightly) so that it is true, and
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Just need help with Questions D and E. Thank you so much.

Transcribed Image Text:4. Let \( m \) and \( b \) be real numbers, and consider the following three functions: \( f(x) = 2x + 1 \), \( g(x) = -\frac{1}{3}x + 2 \), and \( h(x) = mx + b \).
A. If function \( f \) has a codomain of \( (5, 7) \cup (7, 9) \), the largest its domain can be chosen is \( (2, 3) \cup (3, 4) \). Explain.
B. If function \( g \) has a codomain of \( (-1, 1) \cup (1, 3) \), the largest its domain can be chosen is \( (-3, 3) \cup (3, 9) \). Explain.
C. Within the context of the \(\epsilon\)-\(\delta\) definition of a limit, your result from part A suggests that if \(\epsilon\) is equal to 2, the largest \(\delta\) that can be chosen for function \( f(x) \) is 1. Explain.
D. Within the context of the \(\epsilon\)-\(\delta\) definition of a limit, your result from part B suggests that if \(\epsilon\) is equal to 2, the largest \(\delta\) that can be chosen for function \( g(x) \) is 6. Explain.
E. The following statement is (ever so slightly) false: "Within the context of the \(\epsilon\)-\(\delta\) definition of a limit, the results from parts B and C suggest that for a given positive real number \(\epsilon\), the largest that \(\delta\) can be chosen for function \( h(x) \) is \(\frac{\epsilon}{m}\)." Modify this statement (ever so slightly) so that it is true, and briefly describe why the \(\epsilon\)-\(\delta\) definition of a limit demands such a modification be applied.
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