Question

Determine whether the function is one-toone on its entire domain and therefore has an inverse function. $$ f(x)=2-x-x^{3} $$

Short answer

The given function \(f(x)=2-x-x^3\) is one-to-one and therefore has an inverse function.

Step-by-step solution

Find the derivative of the function

Given the function \(f(x)=2-x-x^3\), the first derivative of the function, denoted by \(f'(x)\), can be calculated using the power rule for differentiation, which states that the derivative of \(x^n\) where n is any real number, is \(n*x^{n-1}\). Thus, differentiating the given function, we obtain \(f'(x)=-1-3x^2\).

Analyze the derivative function

Next, the derivative function \(f'(x)=-1-3x^2\) should be analyzed to determine its signs. This derivative is a quadratic function, and it can be observed that \(f'(x)<0\) for all \(x\). Since all values of \(x\) will yield a negative result, the original function is always decreasing.

Determine whether the function is one-to-one

Based on the analysis in Step 2, the function is always decreasing as it does not have any local maxima or minima. Hence, every y-value in the range of the function is associated with a unique x-value in its domain. This means that the function \(f(x)=2-x-x^3\) is one-to-one on its entire domain.

Ascertain if the function has an inverse

A function has an inverse if it's one-to-one or, in other words, when each x-value has a unique y-value. Since the function \(f(x)=2-x-x^3\) is one-to-one, it has an inverse function.