Chapter 1: Q38E (page 35)
Prove the statement using the\(\varepsilon \), \(\delta \)definition of a limit.
\(\mathop {\lim }\limits_{x \to a} c = c\)
It is proved that\(\mathop {\lim }\limits_{x \to a} c = c\).
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Find the functions
(a) \({\bf{f}} \circ {\bf{g}}\)
(b) \({\bf{g}} \circ {\bf{f}}\)
(c) \({\bf{f}} \circ {\bf{f}}\)
(d) \({\bf{g}} \circ {\bf{g}}\)
and their domains.
\({\bf{f}}\left( {\bf{x}} \right){\bf{ = }}{{\bf{x}}^{\bf{2}}}{\bf{ - 1}}\) \({\bf{g}}\left( {\bf{x}} \right){\bf{ = 2x + 1}}\)
Sketch a rough graph of the market value of a new car as a function of time for a period of 20 years. Assume the car is well maintained.
The graph of\({\bf{x = f}}\left( {\bf{x}} \right)\)is given. Match each equation with its graph and give reasons for your choices.
(a)\({\bf{y = f}}\left( {{\bf{x - 4}}} \right)\)
(b)\({\bf{y = f}}\left( {\bf{x}} \right){\bf{ + 3}}\)
(c)\({\bf{y = }}\frac{{\bf{1}}}{{\bf{3}}}{\bf{f}}\left( {\bf{x}} \right)\)
(d)\({\bf{y = - f}}\left( {{\bf{x + 4}}} \right)\)
(e)\({\bf{y = 2f}}\left( {{\bf{x + 6}}} \right)\)
Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations.
\(y = \frac{2}{{x + 1}}\)
Find the functions (a)\({\bf{f}} \circ {\bf{g}}\), (b)\({\bf{g}} \circ {\bf{f}}\), (c)\({\bf{f}} \circ {\bf{f}}\), and (d) \({\bf{g}} \circ {\bf{g}}\) and their domains.
\({\bf{f}}\left( {\bf{x}} \right){\bf{ = x + }}\frac{{\bf{1}}}{{\bf{x}}}\) \({\bf{g}}\left( {\bf{x}} \right){\bf{ = }}\frac{{{\bf{x + 1}}}}{{{\bf{x + 2}}}}\)
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