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Answer:
The domain of a function f(x) is the set of all values for which the function is defined & range of the function is the set of all values that function takes.
Let us take f(x) = 2x+3 or y = 2x+3
The domain of f(x) is (−∞,∞) & the range of f(x) is also (−∞,∞) .
The value of y is described in terms of x . Now, let’s describe x in terms of y :
x = y−32 or g(y) = y−32
The inverse function f−1(x) is a function that "reverses" another function. In f(x) , if you replace x with f−1(x) , then the value of f(f−1(x)) will be equal to x .
In this case, function g is equal to function f−1 .
g(x) = x−32
f−1(x) = x−32
Now coming back to domain and range, the domain of f−1(x) is equal to the range of f(x) & the range of f−1(x) is equal to the domain of f(x) .
This time,
Let us take f(x) = x2
The domain of f(x) is (−∞,∞) but the range of f(x) is (0,∞) .
f−1(x) = x−−√
The domain of f−1(x) is (0,∞) but the range of f−1(x) is (−∞,∞) .
Therefore, if you want to determine the domain and range with inverse functions, then first you have to find out the domain and range of f(x) & then find out the inverse function f−1(x) . The domain of f−1(x) is the range of f(x) & the range of f−1(x) is the domain of f(x) .