**Educational Website Transcription and Explanation:**---### Topic: Inverse Functions and Derivatives**Function Analysis:**The function $ f(x) = 15e^{-x} \sqrt{x} $ is shown below. We are examining its behavior for the restricted domain $ x \geq 0.5 $.**Question 1:** Does the function appear to have an inverse function for this restricted domain? (DO NOT DERIVE IT!!)- **Why or why not?****Question 2:** Estimate $ (f') (1) $: ______- What then is the approximate derivative $ (f^{-1})'(x) $ at the point $\left( \frac{15}{e}, 1 \right)$?- **Why? What theorem or rule supports your answer?****Graph Analysis:**The graph of the function $ f(x) = 15e^{-x} \sqrt{x} $ is provided. Key features include:- The graph decreases as $ x $ increases from 0.5.- The plotted point (0.5, 6.133) is marked, indicating the function value at $ x = 0.5 $.### Detailed Explanation:1. **Does the function have an inverse?** A function has an inverse if it is one-to-one (bijective) in the specified domain. The graph shows a decrease in the value of $ f(x) $ as $ x $ increases from 0.5, suggesting that the function is one-to-one on this domain.2. **Estimating $ f'(1) $:** - Examine the graph or utilize the derivative of the function $ f(x) $ to approximate $ f'(1) $. 3. **Derivative of the Inverse:** - Use the formula for the derivative of the inverse function: \[ (f^{-1})'(y) = \frac{1}{f'(f^{-1}(y))} \] - Understand that this is supported by the Inverse Function Theorem, which states that if $ f $ is bijective and differentiable, then so is $ f^{-1} $.By reviewing the graph and applying these concepts, we can analyze and estimate the behavior of both the function and its inverse.---

f(x)=15xe-x , x≥.5We have to find inverse of this function i.e, f-1(x)We know…

Answered: What the Why? What theorem or rule su

Math

Calculus

What the Why? What theorem or rule su

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

**Educational Website Transcription and Explanation:**

---

### Topic: Inverse Functions and Derivatives

**Function Analysis:**

The function $ f(x) = 15e^{-x} \sqrt{x} $ is shown below. We are examining its behavior for the restricted domain $ x \geq 0.5 $.

**Question 1:** Does the function appear to have an inverse function for this restricted domain? (DO NOT DERIVE IT!!)

- **Why or why not?**

**Question 2:** Estimate $ (f') (1) $: ______

- What then is the approximate derivative $ (f^{-1})'(x) $ at the point $\left( \frac{15}{e}, 1 \right)$?

- **Why? What theorem or rule supports your answer?**

**Graph Analysis:**

The graph of the function $ f(x) = 15e^{-x} \sqrt{x} $ is provided. Key features include:

- The graph decreases as $ x $ increases from 0.5.
- The plotted point (0.5, 6.133) is marked, indicating the function value at $ x = 0.5 $.

### Detailed Explanation:

1. **Does the function have an inverse?**

A function has an inverse if it is one-to-one (bijective) in the specified domain. The graph shows a decrease in the value of $ f(x) $ as $ x $ increases from 0.5, suggesting that the function is one-to-one on this domain.

2. **Estimating $ f'(1) $:**

- Examine the graph or utilize the derivative of the function $ f(x) $ to approximate $ f'(1) $.

3. **Derivative of the Inverse:**

- Use the formula for the derivative of the inverse function:
\[
(f^{-1})'(y) = \frac{1}{f'(f^{-1}(y))}
\]
- Understand that this is supported by the Inverse Function Theorem, which states that if $ f $ is bijective and differentiable, then so is $ f^{-1} $.

By reviewing the graph and applying these concepts, we can analyze and estimate the behavior of both the function and its inverse.

---

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