Questions: For the function F(x)=-2+3/(x-4)^2, (a) graph the rational function using transformations, (b) use the final graph to find the domain and range, and (c) use the final graph to list any vertical, horizontal, or oblique asymptotes. (a) Choose the correct graph below. A. B. C. D. (b) What is the domain of the given function? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The domain of the given function is x x is a real number, x ≠ . (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. The domain of the given function is x x is a real number, x< . (Type an integer or a simplified fraction.) C. The domain of the given function is x x is a real number, x> . (Type an integer or a simplified fraction.) D. The domain of the given function is the set of all real numbers.

For the function F(x)=-2+3/(x-4)^2, (a) graph the rational function using transformations, (b) use the final graph to find the domain and range, and (c) use the final graph to list any vertical, horizontal, or oblique asymptotes. (a) Choose the correct graph below. A. B. C. D. (b) What is the domain of the given function? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The domain of the given function is x x is a real number, x ≠ . (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. The domain of the given function is x x is a real number, x< . (Type an integer or a simplified fraction.) C. The domain of the given function is x x is a real number, x> . (Type an integer or a simplified fraction.) D. The domain of the given function is the set of all real numbers.
Transcript text: For the function $F(x)=-2+\frac{3}{(x-4)^{2}}$, (a) graph the rational function using transformations, (b) use the final graph to find the domain and range, and (c) use the final graph to list any vertical, horizontal, or oblique asymptotes. (a) Choose the correct graph below. A. B. C. D. (b) What is the domain of the given function? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The domain of the given function is $\{x \mid x$ is a real number, $x \neq$ $\square$ \}. (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. The domain of the given function is $\{x \mid x$ is a real number, $x<$ \}. $\square$ (Type an integer or a simplified fraction.) C. The domain of the given function is $\{x \mid x$ is a real number, $x>$ \}. $\square$ (Type an integer or a simplified fraction.) D. The domain of the given function is the set of all real numbers.
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Solution

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Solution Steps

Step 1: Identify Asymptotes

The vertical asymptote is at \(x = 4\). The horizontal asymptote is at \(y = -2\).

Step 2: Determine Domain

The domain of the function is all real numbers except where the denominator equals zero, thus the domain is {x | x ≠ 4}.

Step 3: Determine Range

Given that \(a = 3\), the function approaches the horizontal asymptote \(y = -2\) from above, indicating the range is {y | y > -2}.

Final Answer:

The function \(F(x) = \frac{3}{(x - 4)^{n}} - 2\) has a vertical asymptote at \(x = 4\), a horizontal asymptote at \(y = -2\), a domain of {x | x ≠ 4}, and a range of {y | y > -2}.

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