Questions: f(x)=4/(x+7)

f(x)=4/(x+7)
Transcript text: 22. $f(x)=\frac{4}{x+7}$
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Solution

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Solution Steps

To find the inverse of the function \( f(x) = \frac{4}{x+7} \), we need to swap the roles of \( x \) and \( y \) and solve for \( y \). This involves setting \( y = \frac{4}{x+7} \), then solving for \( x \) in terms of \( y \), and finally expressing \( y \) as a function of \( x \).

Step 1: Set Up the Equation for the Inverse

To find the inverse of the function \( f(x) = \frac{4}{x+7} \), we start by setting \( y = \frac{4}{x+7} \).

Step 2: Swap Variables and Solve for \( x \)

Next, we swap \( x \) and \( y \) to get \( x = \frac{4}{y+7} \). We then solve for \( y \) in terms of \( x \).

Step 3: Solve for \( y \)

Rearranging the equation \( x = \frac{4}{y+7} \), we multiply both sides by \( y+7 \) to get: \[ x(y+7) = 4 \]

Expanding and solving for \( y \), we have: \[ xy + 7x = 4 \] \[ xy = 4 - 7x \] \[ y = \frac{4 - 7x}{x} \]

Final Answer

\(\boxed{f^{-1}(x) = \frac{4 - 7x}{x}}\)

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