Write an equation for the inverse function of each one-to-one function given. ƒ(x) = 3^x

Verified step by step guidance

1. Start with the given function: \(f(x) = 3^x\). To find the inverse function, we first replace \(f(x)\) with \(y\), so we have \(y = 3^x\).
2. Next, interchange the roles of \(x\) and \(y\) to find the inverse. This means we write \(x = 3^y\).
3. To solve for \(y\), apply the logarithm with base 3 to both sides of the equation. This gives \(\log_3(x) = \log_3(3^y)\).
4. Using the logarithmic identity \(\log_b(b^k) = k\), simplify the right side to get \(\log_3(x) = y\).
5. Finally, rewrite \(y\) as the inverse function notation: \(f^{-1}(x) = \log_3(x)\). This is the equation for the inverse function.

Key Concepts

One-to-One Functions

A one-to-one function is a function where each output corresponds to exactly one input, ensuring the function passes the horizontal line test. This property is essential for a function to have an inverse because it guarantees that the inverse will also be a function.

Inverse Functions

An inverse function reverses the effect of the original function, swapping inputs and outputs. If f(x) maps x to y, then its inverse f⁻¹(x) maps y back to x. Finding the inverse involves solving the equation y = f(x) for x in terms of y.

Logarithmic Functions as Inverses of Exponential Functions

The inverse of an exponential function f(x) = a^x (where a > 0 and a ≠ 1) is the logarithmic function with base a, written as f⁻¹(x) = log_a(x). This relationship allows us to express the inverse of 3^x as log base 3 of x.