a) f (2) = 1
b) [tex]f^{-1}(1)=2[/tex]
c) [tex]f^{-1}(f(2))=2[/tex]
Step-by-step explanation:
[tex]f^{-1}[/tex] - Indicates that we have to find the inverse of the function
Given data:
[tex]f(x)=\left(\frac{1}{2}\right) x[/tex] --------> eq.1
[tex]f^{-1}(x)=2 x[/tex] ---------> eq.2
To find [tex]f(2), f^{-1}(1), f^{-1}(f(2))[/tex]
Case a)
Now, substitute x = 2 in the equation 1 to find f (2)
[tex]f(2)=\left(\frac{1}{2}\right) \times 2=1[/tex]
Case b)
Now, substitute x = 1 in the equation 2 to find [tex]f^{-1}(1)[/tex]
[tex]f^{-1}(1)=2(1)=2[/tex]
Case c)
In general,
[tex]f^{-1}(f(x))=f\left(f^{-1}(x)\right)=x[/tex]
Thereby,
[tex]f^{-1}(f(2))=2 \text { where } x=2[/tex]