Question

Find the domain and range of ƒ⁻¹ given.

Answer 1

The function

Replace the function notation f (x ) by y .

Interchange the roles x and y .

Replace the variable y  by the inverse notation f ^-1(x ).

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Definition

Domain: The domain of a function is the set of all possible input values (often the "x" variable), which produce a valid output from a particular function. It is the set of all real numbers for which a function is mathematically defined

Domain of a function f(x) is set of those values of x which will make the function mathematically legal or correct.. certain operations like division by zero , square root of a negative number do not exist in real maths.

So, the domain of the above function is all non negative real numbers.

The domain of is { x ∈ R : x ≥ 0 }.

Definition

Range: The range is the set of all possible output values (usually the variable y, or sometimes expressed as f(x)), which result from using a particular function.

Range set is the corresponding values of the function for different values of x.

Since for all non negative real numbers of x, the function is ≥ - 3.

For example the first domain value is 0, then f ^{- 1}(x) = 2(x)^1/3 - 3 = 2(0)^1/3 - 3 = -3, so the range value starting with - 3.

So, the range of the function is always greater than or equal to - 3.

The range of is { y ∈ R : y ≥ - 3 }.