# Part - 1 Lecture - 7 Chapter 1 Relations and Functions

Success… seems to be connected with action. Successful people keep moving. They make mistakes, but they don’t quit. – **Conrad Hilton**

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Lecture 7 Part 1 Questions from Exercise 1.3, NCERT Exemplar, Board’s Question Bank

Revision of inverse functions (Invertible Functions)

**1.** Consider f: R_+ \rightarrow [4,\infty) given by f(x)=x^2+4. Show that f is invertible with the inverse f^{-1} of f given by f^{-1}(y)=\sqrt{y-4}, where \text{R}_+ is the set of all non-negative real numbers. (N)

**2.** Consider f:\text{R}_+ \rightarrow [-5,\infty) given by f(x)=9x^2+6x-5. Show that f is invertible with f^{-1}(y)=\left ( \frac{\sqrt{y+6}-1}{3}\right) . (N)

**3.** Let f\prime: \text{N} \rightarrow \text{R} be a function defined as f\prime(x)=4x^2+12x+15. Show that f:\text{N} \rightarrow S, where, S is the range of f, is invertible. Find the inverse of f. (N)

**4.** If the function f:\text{R} \rightarrow \text{R} be defined by f(x)=2x-3 and g: \text{R} \rightarrow \text{R} by g(x)=x^3+5, then show that fog is invertible. Also find (fog)^{-1}(x), hence find (fog)^{-1}(9). (B)