Find the inverse of the function \( f(x) = rac{2x + 3}{x - 1} \).
["# How to Find the Inverse of the Function ( f(x) = \frac{2x + 3}{x - 1} ): A Step-by-Step Guide", "Understanding how to find the inverse of a function is essential in algebra and higher mathematics. The inverse function essentially reverses the operation of the original function. If you’re wondering, “what is the inverse of ( f(x) = \frac{2x + 3}{x - 1} )?” — this article provides a clear, step-by-step solution. Whether you’re a student, educator, or self-learner, mastering this method will help you solve complex rational functions and improve your analytical skills.", "---", "## What is a Function Inverse?", "Before diving into calculations, it’s important to grasp the concept. The inverse function, denoted as ( f^{-1}(x) ), satisfies:", "[\nf(f^{-1}(x)) = x \quad \ ext{and} \quad f^{-1}(f(x)) = x\n]", "Finding ( f^{-1}(x) ) means swapping ( x ) and ( y ) in the original equation and solving for ( y ).", "---", "## Step-by-Step: Finding ( f^{-1}(x) ) for ( f(x) = \frac{2x + 3}{x - 1} )", "### Step 1: Replace ( f(x) ) with ( y )", "Start by rewriting the function using ( y ):", "[\ny = \frac{2x + 3}{x - 1}\n]", "### Step 2: Swap ( x ) and ( y )", "To find the inverse, swap ( x ) and ( y ):", "[\nx = \frac{2y + 3}{y - 1}\n]", "### Step 3: Solve for ( y )", "Now isolate ( y ):", "[\nx(y - 1) = 2y + 3\n]", "Distribute ( x ) on the left:", "[\nxy - x = 2y + 3\n]", "Gather all terms containing ( y ) on one side:", "[\nxy - 2y = x + 3\n]", "Factor out ( y ):", "[\ny(x - 2) = x + 3\n]", "Solve for ( y ) by dividing both sides by ( (x - 2) ):", "[\ny = \frac{x + 3}{x - 2}\n]", "### Step 4: Write the inverse function", "Since ( y = f^{-1}(x) ), we conclude:", "[\nf^{-1}(x) = \frac{x + 3}{x - 2}\n]", "---", "## Verification: Confirming That ( f(f^{-1}(x)) = x )", "To ensure correctness, verify by composing the functions:", "[\nf(f^{-1}(x)) = f\left( \frac{x + 3}{x - 2} \right) = \frac{2\left( \frac{x + 3}{x - 2} \right) + 3}{\left( \frac{x + 3}{x - 2} \right) - 1}\n]", "Simplify numerator and denominator:", "Numerator:", "[\n\frac{2(x + 3) + 3(x - 2)}{x - 2} = \frac{2x + 6 + 3x - 6}{x - 2} = \frac{5x}{x - 2}\n]", "Denominator:", "[\n\frac{x + 3 - (x - 2)}{x - 2} = \frac{x + 3 - x + 2}{x - 2} = \frac{5}{x - 2}\n]", "Now divide:", "[\nf(f^{-1}(x)) = \frac{5x/(x - 2)}{5/(x - 2)} = x\n]", "Confirmed: The inverse satisfies the defining property.", "---", "## Why Is This Useful?", "Knowing the inverse function allows you to:", "- Solve equations where ( f(x) = y ) and find ( x = f^{-1}(y) ).\n- Analyze symmetry and behavior shifts between functions and their inverses.\n- Apply transformations systematically in algebra and calculus.", "---", "## Final Answer", "The inverse of the function ( f(x) = \frac{2x + 3}{x - 1} ) is:", "[\n\boxed{ f^{-1}(x) = \frac{x + 3}{x - 2} }\n]", "Learning to find inverses marks a key milestone in understanding function relationships and expands your toolkit for tackling complex mathematical problems.", "---", "Keywords: inverse of function, find inverse function, rational function inverse, step-by-step solve, algebra tutorial, inverse rational function, ( f^{-1}(x) ), function composition, math help.", "---", "For more insights on function inverses, algebraic equations, and practical applications, explore related online resources or consult your textbook—mastery comes through practice and deeper exploration!"]
