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

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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)