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 $\square$

Solution

Solution Steps

Step 1: Explaining the Transformation

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

Step 2: Identifying the Correct Graph

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

Step 3: Finding the Domain

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

Step 4: Finding the Range

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