Question

Let f : R → R + be defined by the formula f(x) = 10^2−x ....

Let f : R → R + be defined by the formula f(x) = 10^2−x . Show that f is injective and surjective, and find the formula for f −1 (x).

Suppose f : A → B and g : B → A. Prove that if f is injective and f ◦ g = iB, then g = f −1 .

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Answer #1

2. f(x) = 102-x

So, f(x) = f(y) implies, 102-x = 102-y

So, 2 - x = 2 - y

So, x = y

Hence, f is injective.

Also, let, y belongs to the co-domain R+, so, y > 0.

Then, f(x) = y implies, 102-x = y

So, 2 - x = log10 y. (Since, y > 0)

So, x = 2 - log10 y belongs to R, such that, f(x) = y

So, f is surjective.

&, f -1(x) = 2 - log10 x

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