Consider the rational function \(f(x) = \frac{2x – 3}{x + 4}\).
(a) Determine the domain of \(f(x)\) and identify any vertical asymptotes.
(b) Find the inverse function \(f^{-1}(x)\) algebraically.
(c) Describe the reflection process used to obtain the graph of \(f^{-1}(x)\) from the graph of \(f(x)\).
Question
Determine the domain of \(f(x) = \frac{2x – 3}{x + 4}\) and identify any vertical asymptotes.
Hint
Hint
Hint
Set the denominator equal to zero to find the exclusion in the domain and the vertical asymptote.
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Answer
Answer
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Explanation
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- Formula Booklet
Complete Answer
The domain is \(\{x \in \mathbb{R} : x
eq -4\}\) with a vertical asymptote at \(x = -4\).
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Answer Explanation
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Question
Find the inverse function \(f^{-1}(x)\) of \(f(x) = \frac{2x – 3}{x+4}\) algebraically.
Hint
Hint
Hint
Swap \(x\) and \(y\) to get \(x = \frac{2y – 3}{y+4}\) and then solve for \(y\).
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Answer
Answer
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Explanation
Explanation
- Formula Booklet
Complete Answer
\(f^{-1}(x)=\frac{3+4x}{2-x}\)
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Question
Describe how you would sketch the graph of \(f^{-1}(x)=\frac{3+4x}{2-x}\) using the reflection of \(f(x)=\frac{2x-3}{x+4}\) about the line \(y=x\).
Hint
Hint
Hint
Remember that the graph of an inverse function is the mirror image of the original function across the line \(y = x\).
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Answer
Answer
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Explanation
Explanation
- Formula Booklet
Complete Answer
Reflect the graph of \(f(x)=\frac{2x-3}{x+4}\) across the line \(y=x\) by swapping each point \((a, b)\) to \((b, a)\). The key features such as asymptotes and intercepts are also reflected accordingly.
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