The domain of the inverse of the function f(x) = 5e^x + 3 is found being all real numbers > 3. Or, we can write this symbolically as: D( f^(-1)(x)) = (3,∞)
What is domain and range of a function?
Domain is the set of values for which the given function is defined.
Range is the set of all values which the given function can output.
What is inverse of a function?
Suppose that the given function is
[tex]f:X\rightarrow Y[/tex]
Then, if function 'f' is one-to-one and onto function (a needed condition for inverses to exist), then, the inverse of the considered function is
[tex]f^{-1}: Y \rightarrow X[/tex]
such that:
[tex]\forall \: x \in X : f(x) \in Y, \exists \: y \in Y : f^{-1}(y) \in X[/tex]
(and vice versa).
It simply means, inverse of 'f' is undo operator, that takes back the effect of 'f'
The domain of the inverse of a function (if it exists) is the range of the function it is inverse of. Similarly range of the inverse of that considered function is the domain of that considered function.
The function given is:
[tex]f(x) = 5e^x + 3[/tex]
The domain of the inverse of this function is this function's range.
Assuming only real values are allowed, for any real number x, we have:
[tex]e^x > 0[/tex]
And therefore, we get:
[tex]5e^x > 0 \: \forall x \in \mathbb R\\5e^x + 3 > 3 \: \forall x \in \mathbb R\\[/tex]
Also, since the function [tex]f(x) = 5e^x + 3[/tex] is strictly monotonically increasing and continuous, it touches all the values as its output from any real number > 3 till any finitely large number.
Thus, its range is (3, ∞) (it is an interval consisting of all the real numbers > 3)
This is the domain of its inverse function.
Thus, the domain of the inverse of the function f(x) = 5e^x + 3 is found being set of all real numbers > 3. Or, we can write this symbolically as: D( f^(-1)(x)) = (3,∞)
Learn more about inverse function here:
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