Exercise 2.3
Question 1
Let S=|1,2.3|. Determine whether the functions $f: S \rightarrow S$ defined as below have inverse. Find $f^{-1}$ if it exists.
(i) $f=\{(1,2),(2,1),(3,1)\}$
(iii) $f=\{(1.1),(2.2),(3.3)\}$
Question 2
State with reason whether following functions have inverse. Also find the inverse if it exists.
(i) $g:(5,6,7,8) \rightarrow\{1,2,3,4\}$ with $g=\{(5,4),(6,3),(7,4),(8.2)\}$
(ii) $f:|1,2,3,4| \rightarrow\{10\}$ with $f=\{(1,10),(2,10) \cdot(3,10),(4,10)\}$
(iii) $h=\{2,3,4.5\} \rightarrow\{7.9 .11 .13\}$ with $h=\{(2.7),(3,9),(4,11),(5,13)\}$
Question 3
Let $f: R \rightarrow R$ be defined by $f(x)=3 x+2$ Show that $f$ is invertible. Find $f^{-1}: R \rightarrow R$.
Question 4
If $f: R \rightarrow R$ is defined by $f(x)=2 x+7$. Prove that $f$ is a bijection. Also. find the inverse of $f$.
Sol :
Question 5
Let R be the set of all real numbers. Show that the function $f: R \rightarrow R$ defined by $f(x)=\frac{3 x+1}{2}$ for all $x \in R$ is bijective. Also find the inverse of $f$.
Question 6
Consider the function $f: R^{+} \rightarrow R^{+}$defined by $f(x)=e^{x} . \forall x \in R^{+}$. where $R^{+}$ denotes the set of all +ve real numbers. Find inverse function of $f$ if it exists.
Sol :
Question 7
If $f:[0, \infty) \rightarrow[2, \infty)$ be defined by $f(x)=x^{2}+2, \forall x \in R$. Then find $f^{-1}$.
Question 8
If $A=\{1,2,3,4\}, B=\{2,4,6,8\}$ and $f: A \rightarrow B$ is given by $f(x)=2 x$, then write $f$ and $f^{-1}$ as a set of ordered pairs.
Question 9
Let $f: R \rightarrow R$ given by $f(x)=4 x+3$. Show that $f$ is invertible. Find the inverse of $f$.
Question 10
Let $f: N \rightarrow Y$ be a function defined as $f(x)=4 x+3$, where $Y=\{y \in N: y=4 x+3$ for some $x \in N\}$. Show that $f$ is invertible. Find the inverse.
Question 11
Let $f: R_{+} \rightarrow(4, \infty)$ be given by $f(x)=x^{2}+4$. Show that $f$ is invertible with the inverse $f^{-1}$ of $f$ given by $f^{-1}(y)=\sqrt{y-4}$.
Question 12
Let $f: N \rightarrow R$ be a function defined as $f(x)=4 x^{2}+12 x+15$. Show that $f: N \rightarrow$ range $f$ is invertible. Find the inverse of $f$.
Question 13
Let $Y=\left\{n^{2}: n \in N\right\} \subset N$, consider $f: N \rightarrow Y$ as $f(n)=n^{2}$. Show that $f$ is invertible. Find the inverse of $f$.
Question 14
Consider $f:\{1,2,3\} \rightarrow\{a, b, c\}$ given by $f(1)=a, f(2)=b$ and $f(3)=c$. Find the inverse $\left(f^{-1}\right)^{-1}$ of $f^{-1}$. Also show that $\left(f^{-1}\right)^{-1}=f$.
Question 15
Let $F: R \rightarrow R$ be defined as $f(x)=10 x+7$.
Find the function $g: R \rightarrow R$ such that gof $=f o g=I_{R}$
Question 16
If the function $f: R \rightarrow R$ be defined by $f(x)=2 x-3$ and $g: R \rightarrow R$ by $g(x)=x^{3}+5$, then find the value of $(f o g)^{-1} x$
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