Finding inverses Find the inverse function.


ƒ(x) = 3x - 4

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the inverse of the function \( f(x) = 3x - 4 \). The inverse function, denoted as \( f^{-1}(x) \), is a function that 'reverses' the effect of \( f(x) \).
2. Step 2: Replace \( f(x) \) with \( y \). So, we have \( y = 3x - 4 \).
3. Step 3: Swap \( x \) and \( y \) to find the inverse. This gives us \( x = 3y - 4 \).
4. Step 4: Solve for \( y \). Add 4 to both sides to get \( x + 4 = 3y \).
5. Step 5: Divide both sides by 3 to isolate \( y \). This gives \( y = \frac{x + 4}{3} \). Thus, the inverse function is \( f^{-1}(x) = \frac{x + 4}{3} \).

Key Concepts

Inverse Functions

An inverse function essentially reverses the effect of the original function. If a function f takes an input x and produces an output y, the inverse function f⁻¹ takes y as input and returns x. For a function to have an inverse, it must be one-to-one, meaning each output is produced by exactly one input.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to isolate variables. To find the inverse of a function, you typically start by replacing f(x) with y, then solve for x in terms of y. This process often requires skills such as adding, subtracting, multiplying, and dividing both sides of the equation.

Function Notation

Function notation is a way to denote functions and their outputs clearly. In this context, f(x) represents the output of the function for a given input x. Understanding function notation is crucial for identifying the original function and correctly expressing its inverse, typically denoted as f⁻¹(x).