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Properties of Inverse Function

Show that iff...

Question

Show that if f and g are defined as and

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Solution

Adding (1) and (2), we obtain

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\int_{0}^{a} f (x) g (x) d x = 2 \int_{0}^{a} f (x) d x

f

g

f (x) = f (a - x)

g (x) + g (a - x) = 4

Q. By using the properties of definite integrals, Show that

\int_{0}^{a} f (x) g (x) d x = 2 \int_{0}^{a} f (x) d x

f

g

f (x) = f (a - x)

g (x) + g (a - x) = 4

Q. Consider f : {1, 2, 3} → {a, b, c} and g : {a, b, c} → {apple, ball, cat} defined as f (1) = a, f (2) = b, f (3) = c, g (a) = apple, g (b) = ball and g (c) = cat. Show that f, g and gof are invertible. Find f⁻¹, g⁻¹ and gof⁻¹ and show that (gof)⁻¹ = f ⁻¹o g⁻¹.

Q. Let f = {(3, 1), (9, 3), (12, 4)} and g = {(1, 3), (3, 3) (4, 9) (5, 9)}. Show that gof and fog are both defined. Also, find fog and gof.

Q. If f : Q → Q, g : Q → Q are two functions defined by f(x) = 2 x and g(x) = x + 2, show that f and g are bijective maps. Verify that (gof)⁻¹ = f⁻¹ og ⁻¹.

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