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

Verified step by step guidance

1. <Step 1: Understand the concept of an inverse function. An inverse function essentially reverses the operation of the original function. For a function \( f(x) \), its inverse \( f^{-1}(x) \) will satisfy the condition \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).>
2. <Step 2: Start by replacing \( f(x) \) with \( y \). So, the equation becomes \( y = 3^x \).>
3. <Step 3: Swap \( x \) and \( y \) to find the inverse. This gives us \( x = 3^y \).>
4. <Step 4: Solve for \( y \) in terms of \( x \). To do this, take the logarithm of both sides. You can use the natural logarithm (ln) or the common logarithm (log). For example, using the natural logarithm, you get \( \ln(x) = \ln(3^y) \).>
5. <Step 5: Apply the logarithmic identity \( \ln(a^b) = b \cdot \ln(a) \) to simplify the equation. This results in \( \ln(x) = y \cdot \ln(3) \). Solve for \( y \) by dividing both sides by \( \ln(3) \), giving \( y = \frac{\ln(x)}{\ln(3)} \). Thus, the inverse function is \( f^{-1}(x) = \frac{\ln(x)}{\ln(3)} \).>

Key Concepts

One-to-One Functions

A one-to-one function is a type of function where each output is produced by exactly one input. This means that no two different inputs can yield the same output. This property is crucial for determining whether a function has an inverse, as only one-to-one functions can be inverted without ambiguity.

Inverse Functions

An inverse function essentially reverses the effect of the original function. If a function ƒ takes an input x and produces an output y, the inverse function ƒ⁻¹ takes y back to x. To find the inverse, we typically swap the roles of x and y in the equation and solve for y.

Exponential Functions

Exponential functions are mathematical expressions in the form ƒ(x) = a^x, where a is a positive constant. The function ƒ(x) = 3^x is an example, where the base is 3. Understanding the properties of exponential functions is essential for finding their inverses, which are logarithmic functions.