Questions: If f(x)=4x+5, find the inverse function using the switch-and-solve method.

Transcript text: If $f(x)=4 x+5$, find the inverse function using the switch-and-solve method.

Solution

Solution Steps

To find the inverse of the function $ f(x) = 4x + 5 $, we use the switch-and-solve method. This involves swapping $ x $ and $ y $ in the equation and then solving for $ y $. The resulting expression for $ y $ will be the inverse function.

Step 1: Define the Original Function

The original function is given by: \[ f(x) = 4x + 5 \]

Step 2: Switch Variables

To find the inverse, we switch the roles of $ x $ and $ y $. This gives us the equation: \[ x = 4y + 5 \]

Step 3: Solve for $ y $

We solve the equation $ x = 4y + 5 $ for $ y $: \[ x - 5 = 4y \] \[ y = \frac{x - 5}{4} \]

Step 4: Simplify the Expression

Simplifying the expression for $ y $, we get: \[ y = \frac{x}{4} - \frac{5}{4} \]

Final Answer

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

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