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6.4-Inverses

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

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6.4-Inverses

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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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