Consider the following statements in respect of the function \(f: R-\left\{\dfrac{3}{5}\right\} \to R-\left\{\dfrac{3}{5}\right\}\) such that \(f(x) = \dfrac{3x+2}{5x-3}\): I. \(f(x)\) is a bijective function. II. \(f^{-1}(x) = f(x)\) Which of the statements given above is/are correct?
Both I and II
Cross-multiplying \(y=\frac{3x+2}{5x-3}\) and solving for \(x\) gives \(x=\frac{3y+2}{5y-3}\), so \(f^{-1}(x)=\frac{3x+2}{5x-3}=f(x)\), proving statement II. Since an explicit inverse exists on the stated domain and codomain, \(f\) is both one-one and onto, so it is bijective, proving statement I. Hence both statements are correct.
Similar Questions
Consider the following statements:
1. A function f : Z → Z, defined by f(x) = x + 1, is one-one as well as onto.
2. A function f : N → N, defined by f(x) = x + 1, is one-one but not onto.
Which of the above statements is/are correct?
For \(x = \frac{{\sqrt \pi }}{2}\) , what is the value of [ho(gof)](x)?
What is [fo(fof)](2) equal to?
The function f(x) = |x| - x 3is
If f(x) = 4x + 1 and g(x) = kx + 2 such that fog(x) = gof(x), then what is the value of k ?
Consider the following relations from \(A\) to \(B\), where \(A = \{1, 3, 5\}\) and \(B = \{2, 4, 6, 8\}\):
I. \(\{(1,2), (3,2), (3,6), (5,8)\}\)
II. \(\{(3,4), (5,8), (1,6), (3,2)\}\)
III. \(\{(1,2), (3,6)\}\)
IV. \(\{(1,6), (3,2), (5,2)\}\)
Which of the above is/are function(s) from \(A\) to \(B\)?
Important Questions from Types of Functions
The number of one-to-one functions from {1, 2, 3} to {1, 2, 3, 4, 5} is
Consider the following statements:
1. A function f : Z → Z, defined by f(x) = x + 1, is one-one as well as onto.
2. A function f : N → N, defined by f(x) = x + 1, is one-one but not onto.
Which of the above statements is/are correct?
For \(x = \frac{{\sqrt \pi }}{2}\) , what is the value of [ho(gof)](x)?
What is [fo(fof)](2) equal to?
If A = {1, 2, 3} and B = {4, 5, 6}, then which of the following is bijective function?
