Questions: The function (f(x)=5x+4) is one-to-one. Find an equation for (f^-1(x)), the inverse function. (f^-1(x)=)

The function (f(x)=5x+4) is one-to-one. Find an equation for (f^-1(x)), the inverse function. (f^-1(x)=)
Transcript text: The function $f(x)=5 x+4$ is one-to-one. Find an equation for $\mathrm{f}^{-1}(x)$, the inverse function. $\mathrm{f}^{-1}(\mathrm{x})=$
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Solution

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Solution Steps

To find the inverse of the function \( f(x) = 5x + 4 \), we need to solve for \( x \) in terms of \( y \) where \( y = f(x) \). Then, we will express \( x \) as a function of \( y \), which will be our inverse function \( f^{-1}(x) \).

Solution Approach
  1. Start with the equation \( y = 5x + 4 \).
  2. Solve for \( x \) in terms of \( y \).
  3. Replace \( y \) with \( x \) to get the inverse function \( f^{-1}(x) \).
Step 1: Define the Function

We start with the function defined as: \[ f(x) = 5x + 4 \]

Step 2: Set Up the Equation

To find the inverse function, we set \( y = f(x) \): \[ y = 5x + 4 \]

Step 3: Solve for \( x \)

Rearranging the equation to solve for \( x \): \[ y - 4 = 5x \] \[ x = \frac{y - 4}{5} \]

Step 4: Express the Inverse Function

Now, we replace \( y \) with \( x \) to express the inverse function: \[ f^{-1}(x) = \frac{x - 4}{5} \]

Final Answer

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

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