Finding inverses Find the inverse function.

ƒ(x) = 3x - 4

Verified step by step guidance

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) $.

Step 2: Replace $ f(x) $ with $ y $. So, we have $ y = 3x - 4 $.

Step 3: Swap $ x $ and $ y $ to find the inverse. This gives us $ x = 3y - 4 $.

Step 4: Solve for $ y $. Add 4 to both sides to get $ x + 4 = 3y $.

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} $.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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).

Recommended video: