Questions: The function f(x)=8/(x-6) is one-to-one. (a) Find the inverse of f. f^(-1)(x)= (b) State the domain and range of f. State the domain of f. Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The domain of f is x (Type an inequality. Use integers or fractions for any numbers in the expression.) B. The domain of f is all real numbers. State the range of f. Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The range of f is y b (Type an inequality. Use integers or fractions for any numbers in the expression.) B. The range of f is all real numbers. (c) State the domain and range of f^(-1). State the domain of f^(-1). Select the correct choice below and, if necessary, fill in the answer box within your choice. (d) Graph f, f^(-1), and y=x on the same set of axes.

The function f(x)=8/(x-6) is one-to-one. (a) Find the inverse of f. f^(-1)(x)= (b) State the domain and range of f. State the domain of f. Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The domain of f is x (Type an inequality. Use integers or fractions for any numbers in the expression.) B. The domain of f is all real numbers. State the range of f. Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The range of f is y b (Type an inequality. Use integers or fractions for any numbers in the expression.) B. The range of f is all real numbers. (c) State the domain and range of f^(-1). State the domain of f^(-1). Select the correct choice below and, if necessary, fill in the answer box within your choice. (d) Graph f, f^(-1), and y=x on the same set of axes.
Transcript text: The function $f(x)=\frac{8}{x-6}$ is one-to-one. (a) Find the inverse of $f$. (b) State the domain and range of f . (c) State the domain and range of $f^{-1}$. (d) Graph $f, f^{-1}$, and $y=x$ on the same set of axes. (a) Find the inverse of $f$. \[ f^{-1}(x)= \] (b) State the domain of f . Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The domain of f is $\{\mathrm{x} \mid \square\}$. (Type an inequality. Use integers or fractions for any numbers in the expression.) B. The domain of $f$ is all real numbers. State the range of $f$. Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The range of $f$ is $\{y \mid$ b $\square$ (Type an inequality. Use integers or fractions for any numbers in the expression.) B. The range of $f$ is all real numbers. (c) State the domain of $f^{-1}$. Select the correct choice below and, if necessary, fill in the answer box within your choice.
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Solution

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Solution Steps

Step 1: Find the inverse of \( f(x) \)

To find the inverse of \( f(x) = \frac{8}{x-6} \), we start by setting \( y = \frac{8}{x-6} \) and then solve for \( x \) in terms of \( y \).

\[ y = \frac{8}{x-6} \]

Multiply both sides by \( x-6 \):

\[ y(x-6) = 8 \]

Solve for \( x \):

\[ yx - 6y = 8 \]

\[ yx = 8 + 6y \]

\[ x = \frac{8 + 6y}{y} \]

Thus, the inverse function is:

\[ f^{-1}(x) = \frac{8 + 6x}{x} \]

Step 2: State the domain and range of \( f \)

The function \( f(x) = \frac{8}{x-6} \) is undefined when \( x = 6 \). Therefore, the domain of \( f \) is all real numbers except 6.

\[ \text{Domain of } f: \{ x \mid x \neq 6 \} \]

The range of \( f \) is all real numbers except 0, because \( \frac{8}{x-6} \) can never be 0.

\[ \text{Range of } f: \{ y \mid y \neq 0 \} \]

Step 3: State the domain and range of \( f^{-1} \)

The inverse function \( f^{-1}(x) = \frac{8 + 6x}{x} \) is undefined when \( x = 0 \). Therefore, the domain of \( f^{-1} \) is all real numbers except 0.

\[ \text{Domain of } f^{-1}: \{ x \mid x \neq 0 \} \]

The range of \( f^{-1} \) is all real numbers except 6, because \( \frac{8 + 6x}{x} \) can never be 6.

\[ \text{Range of } f^{-1}: \{ y \mid y \neq 6 \} \]

Final Answer

\[ f^{-1}(x) = \frac{8 + 6x}{x} \] \[ \text{Domain of } f: \{ x \mid x \neq 6 \} \] \[ \text{Range of } f: \{ y \mid y \neq 0 \} \] \[ \text{Domain of } f^{-1}: \{ x \mid x \neq 0 \} \] \[ \text{Range of } f^{-1}: \{ y \mid y \neq 6 \} \]

{"axisType": 3, "coordSystem": {"xmin": -10, "xmax": 10, "ymin": -10, "ymax": 10}, "commands": ["y = 8/(x-6)", "y = (8 + 6x)/x", "y = x"], "latex_expressions": ["$y = \\frac{8}{x-6}$", "$y = \\frac{8 + 6x}{x}$", "$y = x$"]}

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