Functions and Inverses What's the use? If f exists, we could use it to ...
Functions and Inverses What's the use? If f exists, we could use it to ...
Functions and Inverses What's the use? If f exists, we could use it to ...
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<strong>Functions</strong> <strong>and</strong> <strong>Inverses</strong>Defin<strong>it</strong>ion: A function g is said <strong>to</strong> be <strong>the</strong> inverse of f if:1. domain of f = range of g,2. range of f = domain of g, <strong>and</strong>3. y = f(x) if <strong>and</strong> only if g(y) = x.Notation: f 1 is <strong>the</strong> inverse of f.<strong>What's</strong> <strong>the</strong> <strong>use</strong>? <strong>If</strong> f 1 <strong>exists</strong>, <strong>we</strong> <strong>could</strong> <strong>use</strong> <strong>it</strong> <strong>to</strong> solveequations like f(x) = 0, beca<strong>use</strong> <strong>the</strong>n...Jan 281:47 PM<strong>Functions</strong> <strong>and</strong> <strong>Inverses</strong>Visual Calculus <strong>Inverses</strong> of <strong>Functions</strong> (Overview)Visual Calculus <strong>Inverses</strong> of <strong>Functions</strong> (Flash Tu<strong>to</strong>rial)Jan 282:01 PM1
Examples Decide whe<strong>the</strong>r f is one<strong>to</strong>one. ei<strong>the</strong>r <strong>use</strong> horizontal line test or apply defin<strong>it</strong>ion be sure <strong>to</strong> specify <strong>the</strong> domain!Jan 284:37 PMGiven a graph of y = f(x), <strong>we</strong> can sketch <strong>the</strong> inverse:f 1 (x) = (x 2 2)/5a) Sketch f 1 (x).b) State <strong>the</strong> domain <strong>and</strong> range.c) Find a formula for f 1 (x).Jan 284:59 PM2
Find <strong>the</strong> inverse, f 1 (x).Jan 284:45 PMInverse Trig <strong>Functions</strong>Does f(x) = sin(x) have an inverse?Does cos(x)? Does tan(x)?Jan 291:02 PM3
Important Calc II Skill:(<strong>use</strong>d w/ integration method of trig. subst.)(a) Evaluate cos(sin 1 (x))(b) Evaluate tan(cos 1 (x))Jan 291:06 PMDerivatives of Inverse Trig <strong>Functions</strong>(relies on implic<strong>it</strong> differentiation)Formula: d/dx [sin 1 (x)] = 1/sqrt[1x 2 ]Proof:Caution: <strong>the</strong> derivative,not sin 1 <strong>it</strong>self!Jan 291:22 PM4