Inverse Functions

• Inverse functions reverse the operation of each other; if f(a) = b, then f⁻¹(b) = a
• The inverse of f(x) is denoted as f⁻¹(x), where "-1" indicates the inverse, not an exponent
• Only one-to-one functions have inverses that are also functions and pass both the vertical and horizontal line tests
• The domain of f(x) becomes the range of f⁻¹(x), and the range of f(x) becomes the domain of f⁻¹(x)

Finding the Inverse of a Function Algebraically

• Replace f(x) with y to begin the process
• Switch x and y in the equation
• Solve the equation for y
• Replace y with f⁻¹(x) to denote the inverse function

Example: Find the inverse of f(x) = 2x + 3

• Begin by replacing f(x) with y: y = 2x + 3
• Switch x and y: x = 2y + 3
• Solve for y: x - 3 = 2y, which simplifies to y = (x - 3)/2
• Replace y with f⁻¹(x): f⁻¹(x) = (x - 3)/2

Verifying Inverse Functions

• To confirm that f(x) and g(x) are inverses, demonstrate that f(g(x)) = x and g(f(x)) = x
• This shows each function reverses the effect of the other

Example: Verify that f(x) = 2x + 3 and f⁻¹(x) = (x - 3)/2 are inverses

• Evaluate f(f⁻¹(x)): f(f⁻¹(x)) = 2((x - 3)/2) + 3 = (x - 3) + 3 = x
• Evaluate f⁻¹(f(x)): f⁻¹(f(x)) = ((2x + 3) - 3)/2 = (2x)/2 = x
• Because both compositions result in x, the functions are verified as inverses

Graphing Inverse Functions

• f⁻¹(x)'s graph is a reflection of f(x)'s graph across the line y = x
• An inverse function can be graphed by swapping the x and y coordinates of the original function's points

Example: Graphing f(x) = x² (for x ≥ 0) and its inverse

• For f(x) = x² (with x ≥ 0), the vertex is at (0, 0), and it includes points like (1, 1), (2, 4), and (3, 9)
• The inverse function, f⁻¹(x) = √x, includes the points (0, 0), (1, 1), (4, 2), and (9, 3)
• By plotting these points and reflecting f(x)'s graph over y = x, the graph of f⁻¹(x) can be obtained

Domain Restrictions

• A function's domain sometimes needs to be restricted to ensure its inverse is also a function
• The function f(x) = x² lacks an inverse function across its entire domain, proven by the horizontal line test
• By restricting the domain to x ≥ 0, the inverse becomes f⁻¹(x) = √x, thus creating a valid function

Summary of Key Concepts

• Inverse functions "undo" each other, reversing the operation
• Only one-to-one functions possess inverse functions
• To find the inverse function, switch x and y and solve for y
• The graphs of a function and its inverse are reflections of each other over the line y = x
• Domain restrictions might be needed to guarantee the inverse is a function

Explanation of inverse functions, how to find them algebraically, and how to verify them. Inverse functions "undo" each other, meaning if f(a) = b, then f⁻¹(b) = a. Only one-to-one functions have inverses that are also functions.