Answer :
Answer:
Correct answer:
A. I and II
Step-by-step explanation:
First of all, let us have a look at the steps of finding inverse of a function.
1. Replace y with x and x with y.
2. Solve for y.
3. Replace y with [tex]f^{-1}(x)[/tex]
Given that:
[tex]I.\ y=x \\II.\ y=\dfrac{1}x \\III.\ y=x^2 \\IV.\ y=x^3[/tex]
Now, let us find inverse of each option one by one.
I. y = x, a(x) = x
Replacing y with and x with y:
x = y
x = [tex]a^{-1}(x)[/tex] = [tex]a(x)[/tex] Hence, I is true.
II. [tex]y =\dfrac{1}{x}[/tex]
Replacing y with and x with y:
[tex]x =\dfrac{1}{y}[/tex]
[tex]x=\dfrac{1}{a^{-1}(x)}[/tex]
[tex]\Rightarrow a^{-1}(x) = \dfrac{1}{x}[/tex]
[tex]a^{-1}(x)[/tex] = [tex]a(x)[/tex] Hence, II is true.
III. [tex]y =x^{2}[/tex]
Replacing y with and x with y:
[tex]x =y^{2}\\\Rightarrow y = \sqrt x\\\Rightarrow a^{-1}(x) = \sqrt{x} \ne a(x)[/tex]
Hence, III is not true.
IV. [tex]y =x^{3}[/tex]
Replacing y with and x with y:
[tex]x =y^{3}\\\Rightarrow y = \sqrt[3] x\\\Rightarrow a^{-1}(x) = \sqrt[3]{x} \ne a(x)[/tex]
Hence, IV is not true.
Correct answer:
A. I and II