Questions: A function is given by a formula. Determine whether it is one-to-one. f(x)=x^2-15 Yes, it is one-to-one. No, it is not one-to-one.

Solution Steps

We start with the function given by the formula: \[ f(x) = x^2 - 15 \]

Next, we compute the derivative of the function to analyze its behavior: \[ f'(x) = \frac{d}{dx}(x^2 - 15) = 2x \]

We examine the sign of the derivative \( f'(x) \):

• The derivative \( f'(x) = 2x \) is positive when \( x > 0 \) and negative when \( x < 0 \).
• This indicates that the function is increasing for \( x > 0 \) and decreasing for \( x < 0 \).

Since the function \( f(x) \) is not strictly increasing or strictly decreasing over the entire set of real numbers (it decreases for \( x < 0 \) and increases for \( x > 0 \)), it fails the horizontal line test. Therefore, the function is not one-to-one.

Final Answer