Math Even Odd Notes Solution

Pre-Calculus Even, Odd, One-to-One, and Inverse Functions Even & Odd Functions: Even: Symmetry over the y-axis f(-x) = f(x) (x, y) -> (-x, y) Function will stay the same Odd: Symmetry Around the Origin f(-x) = -f(x) (x, y) -> (-x, -y) Function will have all opposite signs Example 1: The graph of a function defined for x ≥ 0 is given. Complete the graph for x < 0 to make: 1. an even function 2. an odd function Example 2: Determine whether the function is even, odd, or neither. 1. f(x) = 7 - x⁶ f(-x) = 7 - (-x)⁶ = 7 - x⁶ Same terms: EVEN 2. f(x) = x³ + x f(-x) = (-x)³ + (-x) = -x³ - x Terms all have opposite signs: ODD 3. f(x) = 2x - x² f(-x) = 2(-x) - (-x)² = -2x - x² Some terms are the same, some changed signs: NEITHER Definition of One-to-One Functions Passes VLT and HLT No repetition in either variable Passes VLT Fails HLT Repetition in the y values Vertical Line Test VS Horizontal Line Test Passes VLT, fails HLT Example 3: Determine whether f(x) = x² - 3x + 2 is one-to-one. Passes VLT, fails HLT NOT ONE-TO-ONE Steps for Finding Inverses of Functions 1. Replace f(x) with y 2. Switch x and y 3. Solve for y 4. Replace y with f⁻¹(x) and identify domain/range Example 4: Find the inverse of the function. 1. f(x) = (x³ - 7)/6 x = (y³ - 7)/6 6x = y³ - 7 6x + 7 = y³ f⁻¹(x) = ∛(6x + 7) 2. f(x) = √(2x - 1) x = √(2y - 1) x² = 2y - 1 (x² + 1)/2 = y f⁻¹(x) = (x² + 1)/2 3. f(x) = 5/(x - 2) x = 5/(y - 2) x(y - 2) = 5 y - 2 = 5/x y = 5/x + 2 f⁻¹(x) = 5/x + 2 or (5 + 2x)/x Graphs of Functions and Their Inverses The graph of f⁻¹ is obtained by reflecting f over the line y = x. Switch x and y coordinates to obtain graph of inverse.