Questions: A one-to-one function is given. Write an equation for the inverse function. s(x)=4/(x+5) s^-1(x)=

Transcript text: A one-to-one function is given. Write an equation for the inverse function. \[ \begin{array}{r} s(x)=\frac{4}{x+5} \\ s^{-1}(x)= \end{array} \]

Solution

Solution Steps

Step 1: Start with the given function

The given function is: \[ s(x) = \frac{4}{x + 5} \]

Step 2: Replace \( s(x) \) with \( y \)

To find the inverse, we first replace \( s(x) \) with \( y \): \[ y = \frac{4}{x + 5} \]

Step 3: Swap \( x \) and \( y \)

Next, we swap \( x \) and \( y \) to find the inverse function: \[ x = \frac{4}{y + 5} \]

Step 4: Solve for \( y \)

Now, solve for \( y \): \[ x(y + 5) = 4 \] \[ xy + 5x = 4 \] \[ xy = 4 - 5x \] \[ y = \frac{4 - 5x}{x} \]

Step 5: Simplify the expression

Simplify the expression for \( y \): \[ y = \frac{4}{x} - 5 \]

Final Answer

The equation for the inverse function is: \[ \boxed{s^{-1}(x) = \frac{4}{x} - 5} \]

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