To find the inverse, set \( y = rac{2x + 3}{x - 1} \) and solve for \( x \).
["# How to Find the Inverse of ( y = \frac{2x + 3}{x - 1} ): Step-by-Step Solution", "Finding the inverse of a function is a key algebraic skill that helps simplify complex relationships and solve equations in many fields, from engineering to economics. If you’ve ever wondered how to find the inverse of ( y = \frac{2x + 3}{x - 1} ), this guide walks you through the process clearly and concisely.", "## Why Find the Inverse?", "The inverse function "reverses" the original function. If ( y = f(x) ), then ( x = f^{-1}(y) ) allows you to determine the original input from a given output. This is essential when modeling reversible processes or solving equations where the dependent variable is expressed in terms of an expression.", "---", "## Step 1: Replace ( f(x) ) with ( y )", "Start by replacing the functional notation consistently:", "[\ny = \frac{2x + 3}{x - 1}\n]", "---", "## Step 2: Swap ( x ) and ( y )", "To find the inverse, always swap ( x ) and ( y ):", "[\nx = \frac{2y + 3}{y - 1}\n]", "---", "## Step 3: Solve for ( y )", "Now solve this equation for ( y ), which will give you ( f^{-1}(x) ).", "Start with:", "[\nx = \frac{2y + 3}{y - 1}\n]", "Multiply both sides by ( y - 1 ) to eliminate the denominator:", "[\nx(y - 1) = 2y + 3\n]", "Expand the left-hand side:", "[\nxy - x = 2y + 3\n]", "Bring all terms containing ( y ) to one side and constants to the other:", "[\nxy - 2y = x + 3\n]", "Factor out ( y ) on the left:", "[\ny(x - 2) = x + 3\n]", "Now solve for ( y ):", "[\ny = \frac{x + 3}{x - 2}\n]", "---", "## Step 4: Write the Inverse Function", "Thus, the inverse function is:", "[\nf^{-1}(x) = \frac{x + 3}{x - 2}\n]", "---", "## Final Thoughts", "Finding inverses like solving ( y = \frac{2x + 3}{x - 1} ) involves substitution and careful algebra. This process ensures accuracy and clarity, essential habits in mathematical problem solving. Remember, inverses are smart tools for reversing relationships and are widely used in calculus, physics, finance, and more.", "Next time you see a rational function like ( y = \frac{ax + b}{cx + d} ), follow these steps: swap variables, isolate the new variable, and simplify — and you’ll find the inverse in no time!", "---", "### Key Search Terms (Keywords for SEO):\n- find the inverse of a rational function\n- solve ( y = \frac{2x + 3}{x - 1} ) for ( x )\n- inverse of a rational equation\n- step-by-step rational inverse\n- how to find the inverse of a fraction function", "Use these terms when publishing or optimizing your content to rank well in search engines for algebra and function inverse practice."]
