3.7e: Exercises for the reciprocal function

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A: Graph translations of $$\dfrac{1}{x}$$

Exercise $$\PageIndex{A}$$

$$\bigstar$$ Graph the given function. Identify the translations on  $$y = \dfrac{1}{x}$$ used to sketch the graph. Then state the domain and range.

 $$f(x) = \dfrac{1}{x−2}$$ $$f(x) = \dfrac{1}{x+3}$$ $$f(x) = \dfrac{1}{x} + 5$$ $$f(x) = \dfrac{1}{x} − 3$$ $$f(x) = \dfrac{1}{x+1} − 2$$ $$f(x) = \dfrac{1}{x−3} + 4$$
 1. Shift right $$2$$ units; domain: $$(−∞, 2) ∪ (2, ∞)$$; range: $$(−∞, 0) ∪ (0, ∞)$$ 3. Shift up $$5$$ units; domain: $$(−∞, 0) ∪ (0, ∞)$$; range: $$(−∞, 1) ∪ (1, ∞)$$ 5. Shift left $$1$$ unit and down $$2$$ units; domain: $$(−∞, −1) ∪ (−1, ∞)$$; range: $$(−∞, −2) ∪ (−2, ∞)$$

$$\bigstar$$ Use the transformations to graph the following functions.

 $$f ( x ) = - \dfrac { 1 } { x }$$ $$f ( x ) = \dfrac { -1 } { x + 1 } + 2$$ $$f ( x ) = \dfrac {- 1 } { x + 2 }$$ $$f ( x ) = \dfrac { 1 } {- x }$$ $$f ( x ) = \dfrac { 1 } {- x + 2 }$$ $$f ( x ) = \dfrac { 1 } { -x - 1 } + 2$$
 7. Reflect over $$x$$-axis; domain: $$(−∞, 0) ∪ (0, ∞)$$; range: $$(−∞, 0) ∪ (0, ∞)$$ 9. Shift left$$2$$ units, Reflect over $$x$$-axis; domain: $$(−∞, -2) ∪ (-2, ∞)$$; range: $$(−∞, 0) ∪ (0, ∞)$$ 11. Shift left$$2$$ units, Reflect over $$y$$-axis; domain: $$(−∞, -2) ∪ (-2, ∞)$$; range: $$(−∞, 0) ∪ (0, ∞)$$

$$\bigstar$$ Graph using translations of $$\dfrac{1}{x}$$ by first using division to rewrite the function.

 $$f(x) = \dfrac{4x-7}{x-2} \\[6pt]$$ $$f(x) = \dfrac{4-x}{x−3} \\[6pt]$$ $$f(x) = \dfrac{-2x-9}{x+5} \\[6pt]$$ $$f(x) = \dfrac{2x+5}{x+3} \\[6pt]$$ $$f(x) = \dfrac{3-4x}{x-1} \\[6pt]$$ $$f(x) = \dfrac{x+1}{x+2} \\[6pt]$$
 13. $$f(x) = 4+\frac{1}{x-2}$$ Right 2, up 4 15. $$f(x) = -2+\frac{1}{x+5}$$ Left 5,  down 2 17. $$f(x) = -4 - \frac{1}{x-1}$$ Right 1, reflect over x-axis, down 4

$$\bigstar$$ Graph using translations of $$\dfrac{1}{x}$$ by first using division to rewrite the function.

 $$f(x) = \dfrac{x−3}{x+2} \\[6pt]$$ $$f(x) = \dfrac{x+2}{x−4} \\[6pt]$$ $$f(x) = \dfrac{5−x}{x+1} \\[6pt]$$ $$f(x) = \dfrac{x+2}{4−x} \\[6pt]$$ $$f(x) = \dfrac{2x−5}{x+1} \\[6pt]$$ $$f(x) = \dfrac{2x+5}{3−x} \\[6pt]$$
 19. $$f(x) = 1-5\frac{1}{x+2}$$ Left 2, Reflect over x-axis, y -> 5y, up 1 21. $$f(x) = -1+6\frac{1}{x+1}$$ Left 1, y -> 6y, down 1 23. $$f(x) = 2 - 7 \frac{1}{x+1}$$ Left 1, reflect over x-axis, y -> 7y, up 2

B: Construct a graph from a verbal description

Exercise $$\PageIndex{B}$$

$$\bigstar$$ Use the given transformation to graph the function. Note the vertical and horizontal asymptotes.

1. The reciprocal function shifted up two units.
2. The reciprocal function shifted down one unit and left three units.
3. The reciprocal squared function shifted to the right $$2$$ units.
4. The reciprocal squared function shifted down $$2$$ units and right $$1$$ unit.
 31. V.A. $$x=0$$, H.A. $$y=2$$ 33. V.A. $$x=2$$, H.A. $$y=0$$

$$\bigstar$$

3.7e: Exercises for the reciprocal function is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.