3.7e: Exercises for the reciprocal function
- Page ID
- 44351
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.
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- Answers to odd exercises:
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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.
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- Answers to odd exercises:
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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.
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- Answers to odd exercises:
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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.
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- Answers to odd exercises:
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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.
- The reciprocal function shifted up two units.
- The reciprocal function shifted down one unit and left three units.
- The reciprocal squared function shifted to the right \(2\) units.
- The reciprocal squared function shifted down \(2\) units and right \(1\) unit.
- Answers to odd exercises.
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31. V.A. \(x=0\), H.A. \(y=2\)
33. V.A. \(x=2\), H.A. \(y=0\)
\( \bigstar \)