4 5 inverse functions
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4-5 INVERSE FUNCTIONS
Objectives:
1. Find the inverse of a function, if it exists.
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INVERSE EXAMPLEConversion formulas come in pairs, for
example:
These formulas “undo” each other, so they are inverses.
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DEFINITION OF INVERSESTwo functions f and g are inverses if: f(g(x)) = x and g(f(x)) = x
To check if two functions are inverses, perform both compositions and make sure both equal x.
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EXAMPLE 1If and
show that f and g are inverses.
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YOU TRY!Show that and
are inverses.
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INVERSE NOTATIONThe inverse of f is written f -1
f -1(x) is the value of f -1 at x
Note: is not
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FINDING INVERSESThe graph of f -1 is the reflection of f
over the line y = xCan be found by
switching x and y in the ordered pairs.
Find equation of f -1
by switching x and yin the equation andsolving for y.
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EXAMPLE 2Let f(x) = 4 – x2 for x 0.Sketch the graph of f and f -1 (x) Find a rule for f -1 (x)
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YOU TRY!Let g(x) = (x – 4)2 – 1 for x 4.Sketch the graph of g and g -1 (x) Find a rule for g -1 (x)
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EXAMPLE 3Suppose a function f has an inverse.
If f(2) = 3, find:
f -1 (3)
f(f -1(3))
f -1(f(2))
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YOU TRY!Suppose a function g has an inverse.
If g(5) = 1, find:
g -1 (1)
g -1(g(5))
g(g -1(1))
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DO ALL FUNCTIONS HAVE INVERSES?
Reflect the graph of y = x2 over the line y = x.
Is the result a function?
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ONE-TO-ONEOnly functions that are one-to-one
have inverses.One-to-one means each x value has
exactly one y value and each y has exactly one x
Can check using horizontal line test.
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EXAMPLE 4 Which functions are one-to-one?Which have inverses?
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YOU TRY! Is h(x) one-to-one? Does it have an inverse?
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EXAMPLE 5State whether each function has an
inverse. If yes, find f -1 (x) and show f(f -1(x)) = f -1(f(x)) = x
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YOU TRY!Does have an
inverse? If so, find f -1 (x).