Answer :
The domain of f(x) to be [0,∞).
If the domain R, f(x) is not one to one, so its inverse exists not a function.
If we restrict the domain of f(x) then we can define an inverse function.
What is meant by inverse function?
An inverse function, even comprehended as an anti function, is a function that can be reversed into another function. Simply put, if any function " f " takes x to y, then its inverse will take y to x. The inverse function is denoted by f-1 or F-1 if the function exists denoted by ' f ' or ' F '.
In order to have an inverse function, a function must be one to one.
In the case of f(x) = x⁴ we find that f(1) = f(−1) =1.
If f(x) exists not one to one on its implicit domain R.
If we restrict the domain of f(x) to [ 0, ∞) then it does contain an inverse function, then [tex]f^{-1}(y)=\sqrt[4]{y }[/tex]
Let the given function be y = f(x) = x⁴
Taking the square root of both ends, then we get
± √y = x²
Let the real values be x, then
We require the left hand side to be non-negative, so assume the non-negative square root.
Take the square root of both sides of the equation then
±[tex]\sqrt[]{\sqrt{y} }[/tex] = x
⇒ x = ± [tex]\sqrt{\sqrt{y} }[/tex] = ± [tex]\sqrt[4]{y}[/tex]
It does not give us a unique value of x in terms of y.
So the inverse function exists not defined, then x ≥ 0,
i.e., restrict the domain of f(x) to be [0,∞).
If the domain R, f(x) exists not one to one, so its inverse exists not a function.
The domain of f(x) then we can describe an inverse function.
To learn more about inverse function, refer to:
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