The functions in Exercises 11-28 are all one-to-one. For each function, a. Find an equation for f^-1(x), the inverse function. b. Verify that your equation is correct by showing that f(ƒ^-1 (x)) = = x and ƒ^-1 (f(x)) = x. f(x) = (2x +1)/(x-3)

Verified step by step guidance

1. Start by writing the function as an equation with y: \(y = \frac{2x + 1}{x - 3}\).
2. To find the inverse, swap x and y: \(x = \frac{2y + 1}{y - 3}\).
3. Solve this new equation for y. Begin by multiplying both sides by \((y - 3)\) to eliminate the denominator: \(x(y - 3) = 2y + 1\).
4. Distribute x on the left side: \(xy - 3x = 2y + 1\). Then, collect all terms involving y on one side and constants on the other: \(xy - 2y = 3x + 1\).
5. Factor y out on the left side: \(y(x - 2) = 3x + 1\). Finally, solve for y by dividing both sides by \((x - 2)\): \(y = \frac{3x + 1}{x - 2}\). This y represents the inverse function \(f^{-1}(x)\).

Key Concepts

One-to-One Functions

A function is one-to-one if each output corresponds to exactly one input, meaning it passes the horizontal line test. This property ensures the function has an inverse because no two different inputs produce the same output, allowing the inverse to be well-defined.

Finding the Inverse Function

To find the inverse of a function, swap the roles of x and y in the equation and solve for y. This process reverses the input-output relationship, producing a function that 'undoes' the original function's operation.

Verification of Inverse Functions

To verify an inverse function, compose the original function with its inverse in both orders: f(f⁻¹(x)) and f⁻¹(f(x)). Both compositions should simplify to x, confirming that the functions are true inverses of each other.