Questions: Explain how the graph of the function f(x) = 1/x + 7 can be obtained from the graph of y = 1/x. Then graph f and give domain and range. To obtain the graph of f, shift the graph of y = 1/x up 7 unit(s). Graph the function f(x) = 1/x + 7. The domain of f(x) is (-∞, 0) U (0, ∞) The range of f(x) is

Explain how the graph of the function f(x) = 1/x + 7 can be obtained from the graph of y = 1/x. Then graph f and give domain and range.

To obtain the graph of f, shift the graph of y = 1/x up 7 unit(s).
Graph the function f(x) = 1/x + 7.
The domain of f(x) is (-∞, 0) U (0, ∞)
The range of f(x) is
Transcript text: Explain how the graph of the function $f(x)=\frac{1}{x}+7$ can be obtained from the graph of $y=\frac{1}{x}$. Then graph $f$ and give domain and range. To obtain the graph of f , shift the graph of $\mathrm{y}=\frac{1}{\mathrm{x}} \quad$ up $\quad 7$ unit(s). Graph the function $f(x)=\frac{1}{x}+7$. The domain of $f(x)$ is $(-\infty, 0) U(0, \infty)$ The range of $f(x)$ is $\square$
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Solution

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Solution Steps

Step 1: Explaining the Transformation

The graph of f(x)=1x+7f(x) = \frac{1}{x} + 7 can be obtained from the graph of y=1xy = \frac{1}{x} by shifting the graph of y=1xy = \frac{1}{x} up by 7 units. The "+ 7" outside the 1x\frac{1}{x} term represents a vertical translation.

Step 2: Identifying the Correct Graph

Since the graph of y=1xy=\frac{1}{x} is shifted upwards by 7 units, the correct graph is B.

Step 3: Finding the Domain

The function f(x)=1x+7f(x) = \frac{1}{x} + 7 is undefined when x=0x=0. Therefore, the domain of f(x)f(x) is (,0)(0,)(-\infty, 0) \cup (0, \infty).

Step 4: Finding the Range

The range of y=1xy = \frac{1}{x} is all real numbers except y=0y=0. Shifting the graph vertically by 7 units shifts the horizontal asymptote up to y=7y=7. Therefore, the range of f(x)f(x) is (,7)(7,)(-\infty, 7) \cup (7, \infty).

Final Answer:

  • Transformation: Shift the graph of y=1xy=\frac{1}{x} up 7 units.
  • Graph: B
  • Domain: (,0)(0,)(-\infty, 0) \cup (0, \infty)
  • Range: (,7)(7,)(-\infty, 7) \cup (7, \infty)
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