Master The Domain And Range Of Inverse Functions With F⁻¹(X) = -3+³√(X+4)

f⁻¹(x)= -3+³√(x+4)

f(x)= (x+3)³-4

To understand how to find the domain and range of a function given in the form of f⁻¹(x), we need to first understand the concept of inverse functions. The inverse of a function essentially swaps the input and output values of the function. For example, if f(x) = y, then the inverse of f(x), denoted by f⁻¹(y), would give us the value of x if we know the value of y.

Now, let’s analyze the given function f⁻¹(x)= -3+³√(x+4) and find its domain and range.

Domain:

The domain of a function gives us all the possible inputs for which the function is defined. In the case of an inverse function, the domain of the inverse function represents all possible outputs of the original function.

In this function, we can see that we have a cube root function within the parentheses. The domain of a cube root function is all real numbers, as cube roots can be found for all real numbers. However, we also have an additional constraint of (x+4) being non-negative, as we cannot find the cube root of a negative number. Therefore, to find the domain of this inverse function, we need to solve the inequality:

x + 4 ≥ 0

x ≥ -4

Thus, the domain of f⁻¹(x) is all real numbers greater than or equal to -4.

Range:

The range of a function represents all possible output values. In the case of an inverse function, the range of the inverse function represents all possible inputs of the original function.

Looking at the function f⁻¹(x)=-3+³√(x+4), we can see that the cube root part of the function will always be non-negative as the domain of our function is restricted to non-negative values. This means that the range of the cube root function is also non-negative.

Therefore, the range of the inverse function f⁻¹(x) is all real numbers greater than or equal to -3.

So, to summarize:

Domain of f⁻¹(x) = {x | x ≥ -4}
Range of f⁻¹(x) = {y | y ≥ -3}

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