Questions: Is the function (f(x)=3 x^3+3) a one-to-one function? Does it have an inverse? Is the function one-to-one? No Yes Does it have an inverse? No Yes

Solution Steps

To determine if the function \( f(x) = 3x^3 + 3 \) is one-to-one, we need to check if it is strictly increasing or decreasing. A function is one-to-one if it passes the horizontal line test, which can be verified by checking if its derivative is always positive or always negative. If the function is one-to-one, it will have an inverse.

A function is one-to-one if every horizontal line intersects the graph of the function at most once. Alternatively, a function is one-to-one if it passes the horizontal line test. Another way to determine if a function is one-to-one is to check if its derivative is always positive or always negative (i.e., the function is strictly increasing or strictly decreasing).

Given the function \( f(x) = 3x^3 + 3 \), we find its derivative:

\[ f'(x) = \frac{d}{dx}(3x^3 + 3) = 9x^2 \]

The derivative \( f'(x) = 9x^2 \) is always non-negative and equals zero only when \( x = 0 \). Since \( f'(x) \) is not strictly positive or strictly negative for all \( x \), the function is not strictly increasing or decreasing. Therefore, the function is not one-to-one.

A function has an inverse if and only if it is one-to-one. Since we determined in Step 1 that the function \( f(x) = 3x^3 + 3 \) is not one-to-one, it does not have an inverse.

Final Answer