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Proving functions are inverses of each other: are these inverses of each other band can you explain why or why not.

Proving functions are inverses of each other: are these inverses of each other band-example-1

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Answer:

a) J(K(x)) = x; K(J(x)) = x; functions are inverses

b) f(g(x)) = 8 -x; g(f(x)) = -x; functions are not inverses

c) f(x) = 4 -x; f^-1(x) = 4 -x. g(x) = x -4; g^-1(x) = x +4

Explanation:

a) Substitute for the function argument in the usual way. If the functions are inverses, their composite is the identity function.

J(K(x)) = J(1/3x -2) = 3(1/3x -2) +6 = x -6 +6 = x . . . . functions are inverses

K(J(x)) = K(3x +6) = 1/3(3x +6) -2 = x +2 -2 = x . . . . functions are inverses

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b) f(g(x)) = f(x -4) = 4 -(x -4) = 4 -x +4 = 8 -x . . . . functions are not inverses

g(f(x)) = g(4 -x) = (4 -x) -4 = -x . . . . functions are not inverses

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c) To find the inverse function for y = f(x), solve x = f(y).

The inverse of f(x) = 4 -x is ...

x = f(y) = 4 -y

y = 4 -x . . . . . add y-x to both sides

f^-1(x) = 4 -x

and for g(x) = x -4, the inverse is ...

x = g(y) = y -4

x +4 = y

g^-1(x) = x +4

User Y Durga Prasad
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