Respuesta :
Answer:
The function can have its own inverse, that is, [tex]$f^{-1}(x)=f(x)$[/tex] and this type of function is called an involution. Example f(x)=6-x.
Step-by-step explanation:
A statement that a function can be its own inverse is given.
It is required to explain whether a function has its own inverse. Then explain whether the given statement satisfies the condition.
Step 1 of 3
Consider a function is f(x)=6-x.
This function is a continuous function for all values of x. This function is also a linear function. So, every continuous linear function is a one-to-one function.
So, this function is one-to-one.
Step 2 of 3
Consider f(x) as y and rewrite the equation.
The equation becomes [tex]$y=6-x$[/tex]
Solve the rewritten equation.
Add -6 on both sides of the equation.
[tex]$\begin{aligned}&y=6-x \\&y-6=6-x-6 \\&y-6=-x\end{aligned}$[/tex]
Step 3 of 3
Multiply by -1 on both sides.
[tex]$\begin{aligned}&y-6=-x \\&(-1)(y-6)=(-x)(-1) \\&-y+6=x \\&x=6-y\end{aligned}$[/tex]
Interchange x and y in solved equation.
[tex]$\begin{aligned}&x=6-y \\&y=6-x\end{aligned}$[/tex]
So, the inverse of the given function is [tex]$f^{-1}(x)=6-x$[/tex].
The function and inverse of the function are the same, that is, [tex]$f^{-1}(x)=f(x)$[/tex]
So, a function can have its own inverse. This type of function is called an involution.
