• NCERT Chapter
Question 7

Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.

fR → R is given by,

f(x) = 4x + 3

One-one:

Let f(x) = f(y).

∴ f is a one-one function.

Onto:

For y ∈ R, let y = 4x + 3.

Therefore, for any y ∈ R,   such that

∴ f is onto.

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

Let us define gR→ 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:- Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2): L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.
• Q:-

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.

• 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}.
• 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.
• Q:-

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.