# 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

Booklets/Notes/Assignments are typed here on this website and their PDFs will be made available soon.

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)