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Beyond the Basics

97. Stretch the graph of $y = \frac{1}{x}$ $y = \frac{1}{x}$ vertically by a factor of 2 and reflect the graph about the x-axis.
99. Shift the graph of $y = \frac{1}{x^{2}}$ $y = \frac{1}{x^{2}}$ two units to the right.
101. Shift the graph of $y = \frac{1}{x^{2}}$ $y = \frac{1}{x^{2}}$ one unit to the right and two units down.
103. Shift the graph of $y = \frac{1}{x^{2}}$ $y = \frac{1}{x^{2}}$ six units to the left.
105. Shift the graph of $y = \frac{1}{x^{2}}$ $y = \frac{1}{x^{2}}$ one unit to the right and one unit up.
107.
1. $f (x) \to 0$ $f (x) \to 0$ as $x \to - \infty, f (x) \to 0$ $x \to - \infty, f (x) \to 0$ as $x \to \infty, f (x) \to - \infty$ $x \to \infty, f (x) \to - \infty$ as $x \to 0^{-},$ $x \to 0^{-},$ and $f (x) \to \infty$ $f (x) \to \infty$ as $x \to 0^{+};$ $x \to 0^{+};$ no x-intercept; no y-intercept; vertical asymptote: y-axis; horizontal asymptote: x-axis. The graph is above the x-axis on $(0, \infty)$ $(0, \infty)$ and below the x-axis on $(- \infty, 0) .$ $(- \infty, 0) .$
2. $f (x) \to 0$ $f (x) \to 0$ as $x \to - \infty, f (x) \to 0$ $x \to - \infty, f (x) \to 0$ as $x \to \infty, f (x) \to \infty$ $x \to \infty, f (x) \to \infty$ as $x \to 0^{-},$ $x \to 0^{-},$ and $f (x) \to - \infty$ $f (x) \to - \infty$ as $x \to 0^{+};$ $x \to 0^{+};$ no x-intercept; no y-intercept; vertical asymptote: y-axis; horizontal asymptote: x-axis. The graph is above the x-axis on $(- \infty, 0)$ $(- \infty, 0)$ and below the x-axis on $(0, \infty) .$ $(0, \infty) .$
3. $f (x) \to 0$ $f (x) \to 0$ as $x \to \infty, f (x) \to 0$ $x \to \infty, f (x) \to 0$ as $x \to - \infty, f (x) \to \infty$ $x \to - \infty, f (x) \to \infty$ as $x \to 0^{-},$ $x \to 0^{-},$ and $f (x) \to \infty$ $f (x) \to \infty$ as $x \to 0^{+};$ $x \to 0^{+};$ no x-intercept; no y-intercept; vertical asymptote: y-axis; horizontal asymptote: x-axis. The graph is above the x-axis on $(- \infty, 0) \cup (0, \infty) .$ $(- \infty, 0) \cup (0, \infty) .$
4. $f (x) \to 0$ $f (x) \to 0$ as $x \to - \infty, f (x) \to 0$ $x \to - \infty, f (x) \to 0$ as $x \to \infty, f (x) \to - \infty$ $x \to \infty, f (x) \to - \infty$ as $x \to 0^{-},$ $x \to 0^{-},$ and $f (x) \to - \infty$ $f (x) \to - \infty$ as $x \to 0^{+};$ $x \to 0^{+};$ no x-intercept; no y-intercept; vertical asymptote: y-axis; horizontal asymptote: x-axis. The graph is below the x-axis on $(- \infty, 0) \cup (0, \infty) .$ $(- \infty, 0) \cup (0, \infty) .$
109.
111. ${[f (x)]}^{- 1} = \frac{1}{2 x + 3}$ ${[f (x)]}^{- 1} = \frac{1}{2 x + 3}$ and $f^{- 1} (x) = \frac{1}{2} x - \frac{3}{2} .$ $f^{- 1} (x) = \frac{1}{2} x - \frac{3}{2} .$ The two functions are different.
113. g(x) has the oblique asymptote $y = 2 x + 3; y \to - \infty$ $y = 2 x + 3; y \to - \infty$ as $x \to - \infty,$ $x \to - \infty,$ and $y \to \infty$ $y \to \infty$ as $x \to \infty .$ $x \to \infty .$
115. Point of intersection: $(- \frac{1}{3}, 1);$ $(- \frac{1}{3}, 1);$ horizontal asymptote $y = 1$ $y = 1$
117. $f (x) = \frac{2 - x}{x - 3}$ $f (x) = \frac{2 - x}{x - 3}$
119. $f (x) = \frac{- x^{2} + 1}{x}$ $f (x) = \frac{- x^{2} + 1}{x}$
121. $f (x) = (x + 4) / (x - 2)$ $f (x) = (x + 4) / (x - 2)$
123. $f (x) = (4 x^{2} - 1) / {(x - 1)}^{2}$ $f (x) = (4 x^{2} - 1) / {(x - 1)}^{2}$
125. $f (x) = (3 x^{2} - x + 1) / (x - 1)$ $f (x) = (3 x^{2} - x + 1) / (x - 1)$