Questions: Determine whether the function is one-to-one. If it is, find a formula for its inverse. f(x)=x^3-5 Is the function one-to-one? Yes No Select the correct choice below and fill in any answer boxes within your choice. A. The inverse function is f^(-1)(x)= B. There is no inverse function.

Solution Steps

To determine if the function \( f(x) = x^3 - 5 \) 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 the derivative. If the derivative is always positive or always negative, the function is one-to-one. If the function is one-to-one, we can find its inverse by solving the equation \( y = x^3 - 5 \) for \( x \).

A function is one-to-one if each output is produced by exactly one input. For a function \( f(x) = x^3 - 5 \), we can determine if it is one-to-one by checking if it is strictly increasing or decreasing.

The derivative of \( f(x) \) is: \[ f'(x) = \frac{d}{dx}(x^3 - 5) = 3x^2 \]

Since \( 3x^2 \geq 0 \) for all \( x \) and \( f'(x) = 0 \) only at \( x = 0 \), the function is non-decreasing and strictly increasing for \( x \neq 0 \). Therefore, \( f(x) = x^3 - 5 \) is a one-to-one function.

To find the inverse function, we need to solve the equation \( y = x^3 - 5 \) for \( x \).

1. Start with the equation: \[ y = x^3 - 5 \]

2. Add 5 to both sides: \[ y + 5 = x^3 \]

3. Take the cube root of both sides: \[ x = \sqrt[3]{y + 5} \]

Thus, the inverse function is: \[ f^{-1}(x) = \sqrt[3]{x + 5} \]

Final Answer