• NCERT Chapter
Question 8

# Consider f : R+ → [4, ∞) given by f(x) = x2 + 4. Show that f is invertible with the inverse f–1 of f given by , where R+ is the set of all non-negative real numbers.

f : R+ → [4, ∞) is given as f(x) = x2 + 4.

One-one:

Let f(x) = f(y).

∴ f is a one-one function.

Onto:

For y ∈ [4, ∞), let y = x2 + 4.

Therefore, for any y ∈ R, there exists  such that

.

∴ f is onto.

Thus, f is one-one and onto and therefore, f - 1 exists.

Let us define g: [4, ∞)  → R+ by,

Hence, f is invertible and the inverse of f is given by

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Let A = R – {3} and B = R – {1}. Consider the function  f : A → B defined by

• Q:- Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = { (a,b) ; |a - b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.
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Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

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Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.

• Q:- Show that the relation R in the set R of real numbers, defined as R = {(a, b): a ≤ b2} is neither reflexive nor symmetric nor transitive.
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