Write an equation for the inverse function of each one-to-one function given. ƒ(x) = 4^x+2

Verified step by step guidance

1. Start with the given function: \(f(x) = 4^{x} + 2\).
2. To find the inverse function, first replace \(f(x)\) with \(y\): \(y = 4^{x} + 2\).
3. Swap the variables \(x\) and \(y\) to reflect the inverse relationship: \(x = 4^{y} + 2\).
4. Isolate the exponential term by subtracting 2 from both sides: \(x - 2 = 4^{y}\).
5. Take the logarithm base 4 of both sides to solve for \(y\): \(y = \log_{4}(x - 2)\), which gives the inverse function \(f^{-1}(x) = \log_{4}(x - 2)\).

Key Concepts

One-to-One Functions

A one-to-one function is a function where each output corresponds to exactly one input, ensuring it has an inverse. This property is essential because only one-to-one functions have inverses that are also functions. Verifying this helps confirm that the inverse function exists.

Inverse Functions

An inverse function reverses the effect of the original function, swapping inputs and outputs. To find the inverse, you replace f(x) with y, interchange x and y, and then solve for y. The inverse function essentially 'undoes' the original function's operation.

Exponential and Logarithmic Functions

Since the given function involves an exponential expression (4^(x+2)), its inverse will involve logarithms. Understanding that logarithms are the inverses of exponential functions is crucial for solving for the inverse function. Specifically, the inverse uses the logarithm base 4 to isolate the variable.