Questions: For the following exercise, find (f^-1(x)) for the function. (f(x)=6-x) (f^-1(x)=)

For the following exercise, find (f^-1(x)) for the function. (f(x)=6-x) (f^-1(x)=)
Transcript text: For the following exercise, find $f^{-1}(x)$ for the function. \[ \begin{array}{l} f(x)=6-x \\ f^{-1}(x)= \end{array} \]
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Solution

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Solution Steps

To find the inverse of a function \( f(x) = 6 - x \), we need to swap the roles of \( x \) and \( y \) in the equation and solve for \( y \). This involves setting \( y = 6 - x \), then solving for \( x \) in terms of \( y \), and finally expressing \( x \) as a function of \( y \).

Step 1: Identify the Function

The given function is \( f(x) = 6 - x \).

Step 2: Swap Variables

To find the inverse, we swap \( x \) and \( y \) in the equation. Start with \( y = 6 - x \) and swap to get \( x = 6 - y \).

Step 3: Solve for the New Variable

Solve the equation \( x = 6 - y \) for \( y \): \[ y = 6 - x \]

Step 4: Express the Inverse Function

The inverse function is expressed as \( f^{-1}(x) = 6 - x \).

Final Answer

The inverse of the function is \(\boxed{f^{-1}(x) = 6 - x}\).

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