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
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Chapter2: Second-order Linear Odes
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Just need help with Questions D and E. Thank you so much. 

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.
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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