\[ f(x)=60.89-x^{2} \] The lead engineer is Ashton. This is a quiz or exam question and is © 2024 MF-LM-JMM. Sharing, posting, copying, saving (etc) this problem in any way or sending/sharing it to any app/website/Al-tool is a violation of the Honor Code. A) Follow all course work guidelines and note any other specific directions below. (6090) What did Ashton find to be the inverse? (FZHI) You must show all steps of your algebraic work to receive full credit. Answer: \( f^{-1}(x)=\square \)

To find the inverse of the function \( f(x) = 60.89 - x^2 \), we start by replacing \( f(x) \) with \( y \): 1. Set \( y = 60.89 - x^2 \). 2. Swap \( x \) and \( y \): \( x = 60.89 - y^2 \). 3. Rearrange to solve for \( y^2 \): \( y^2 = 60.89 - x \). 4. Take the square root of both sides (considering only the principal root for a function): \( y = \sqrt{60.89 - x} \). Thus, the inverse function is \( f^{-1}(x) = \sqrt{60.89 - x} \). --- In the world of engineering, understanding the inverse of functions is crucial for problem-solving, especially when dealing with measurements and outputs. When designers create models, often they need to derive inputs from outputs to ensure their systems meet desired specifications, making inverse functions invaluable! Moreover, a common mistake when finding inverses, particularly with quadratic functions, is forgetting to restrict the domain or considering only one branch of the square root. Without this, you could end up with a function that doesn’t behave like a true inverse, leading to confusion and errors in calculations. Always double-check the validity of your inverse!