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Understanding Inverses: Functions, Relations, and Line Tests

Dive into the concept of inverses in mathematics! This guide covers how to reflect points over the line y=x, the significance of switching x and y values, and the differences between inverse relations and functions. Learn about the vertical and horizontal line tests to determine if graphs represent functions and their inverses. Discover practical examples, including how to verify if two functions are inverses by checking if f(g(x)) = x and g(f(x)) = x. Perfect for visual learners and those seeking clarity in function behavior!

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Understanding Inverses: Functions, Relations, and Line Tests

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  1. 6.4-Inverses • Inverse: • 1) Reflect over the line y=x (y-int.=0, m=1) • 2.) All points have values switched (x,y) → (y,x) • 3.) Equation has x & y switched, & solved for x Opposite operations exist (+ vs. - : x vs. ÷ ) • 4.) Composites = x BOTH ways f(g(x)) = x AND g(f(x)) = x also written as f( • 5.) Symbol for inverse is

  2. Inverse Relations vs. Functions • Inverse Relation: • 1) all points switched (reflect over y=x) • 2) new picture may PASS or FAIL vertical line test • 3) original picture may PASS or FAIL horizontal line test • Inverse Function: • 1) all points switched (reflect over y=x) • 2) new picture PASSES vertical line test • 3) original picture PASSES horizontal line test

  3. Line Tests • Vertical Line Test : • Tests to see if that figure is a FUNCTION • No vertical line crosses graph more than 1 time • Horizontal Line Test: • Tests to see if the inverse WILL be a function • If original passes, then INVERSE will be a function • No horizontal line crosses graph more than once

  4. Find the Inverses • 1. (0,1)(1,2)(2,5) • 2. (0,0) (1,1) (4,2) Graph inverse. • 3. f(x) = 2x-1

  5. Verify 2 Equations are Inverses • Find f(g(x)) • Find g(f(x) • When simplified BOTH = x • Example: • 4. Verify (show, prove) f and g are inverses • f(x) = 4x+9 g(x) = x - OR

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