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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