Question

Find the inverse of the functions. $f(x)=\frac{3 x+4}{5-4 x}$

   Find the inverse of the functions.
$f(x)=\frac{3 x+4}{5-4 x}$

Precalculus

Jay Abramson 1st Edition

Chapter 3, Problem 569

Instant Answer

Solved by Expert Amrita Bhasin

02/25/2020

Step 1

So, $f(x)=\frac{3 x+4}{5-4 x}$ becomes $x=\frac{3 y+4}{5-4 y}$. Show more…

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Find the inverse of the functions. $f(x)=\frac{3 x+4}{5-4 x}$

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Key Concepts

One-to-One Functions

An inverse function exists only if the original function is one-to-one, meaning that each output is produced by a unique input. This property, also known as injectivity, ensures that the inverse mapping properly assigns each element of the range back to one unique element in the domain.

Swapping Variables Method

Finding the inverse of a function typically involves replacing f(x) with y, interchanging the roles of x and y, and then solving the resulting equation for y. This process effectively reverses the function's input-output relationship.

Inverse Function

An inverse function reverses the effect of the original function. For any function f, its inverse f?¹ satisfies the condition f(f?¹(x)) = x and f?¹(f(x)) = x for all x in their respective domains, essentially undoing the action of f.

Algebraic Manipulation

The step-by-step process of solving for the new variable after swapping involves algebraic manipulation techniques such as cross-multiplication, isolating terms, and simplifying expressions. Mastery of these techniques is essential for accurately determining the inverse function.

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