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

Successâ€¦ seems to be connected with action. Successful people keep moving. They make mistakes, but they donâ€™t quit. – **Conrad Hilton**

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Lecture 7 Part 1 Questions from Exercise 1.3, NCERT Exemplar, Boardâ€™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)