
Q1 Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5.
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In the previous class, introduction of limits and derivatives was given. That was basically a calculus introduction. This chapter is a continuation of it. We will study about differentiation of functions. New functions like exponential and logarithmic functions will be introduced. This chapter consists of continuity and differentiability. derivative of a composite function, chain rule, derivatives of inverse trigonometric functions and implicit functions, logarithmic differentiation, parametric forms of derivative of functions, second order derivatives, Rolle's and Lagrange's Mean value theorems.
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Q1  Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5. 
Ans:  Our Experts will give the answer soon. 
Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither oneone nor onto, where [x] denotes the greatest integer less than or equal to x.
Check the injectivity and surjectivity of the following functions:
(i) f : N → N given by f(x) = x^{2}
(ii) f : Z → Z given by f(x) = x^{2}
(iii) f : R → R given by f(x) = x^{2}
(iv) f : N → N given by f(x) = x^{3}
(v) f : Z → Z given by f(x) = x^{3 }
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.
Show that f : [–1, 1] → R, given by is oneone. Find the inverse of the function f : [–1, 1] → Range f.
(Hint: For y ∈ Range f, y =, for some x in [  1, 1], i.e.,)
Consider f : {1, 2, 3} → {a, b, c} given by f(1) = a, f(2) = b and f(3) = c. Find f^{ –1} and show that (f^{ –1})^{–1} = f.
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.