https://file.aiquickdraw.com/mathbot/file/1726391704042obd2ib8b.jpg

Let's find the inverses of each of the given functions step by step.

### 1. \( f(x) = x^3 - 15 \)

To find the inverse, follow these steps:
1. Replace \( f(x) \) with \( y \):  
   \( y = x^3 - 15 \)
2. Swap \( x \) and \( y \):  
   \( x = y^3 - 15 \)
3. Solve for \( y \):  
   \( x + 15 = y^3 \)  
   \( y = \sqrt[3]{x + 15} \)
   
Thus, the inverse of \( f(x) \) is:  
\( f^{-1}(x) = \sqrt[3]{x + 15} \)

---

### 2. \( g(x) = \frac{2x}{3} \)

1. Replace \( g(x) \) with \( y \):  
   \( y = \frac{2x}{3} \)
2. Swap \( x \) and \( y \):  
   \( x = \frac{2y}{3} \)
3. Solve for \( y \):  
   \( y = \frac{3x}{2} \)

Thus, the inverse of \( g(x) \) is:  
\( g^{-1}(x) = \frac{3x}{2} \)

---

### 3. \( h(x) = \frac{x^2}{5} \)

1. Replace \( h(x) \) with \( y \):  
   \( y = \frac{x^2}{5} \)
2. Swap \( x \) and \( y \):  
   \( x = \frac{y^2}{5} \)
3. Solve for \( y \):  
   \( y = \pm \sqrt{5x} \)

Thus, the inverse of \( h(x) \) is:  
\( h^{-1}(x) = \pm \sqrt{5x} \)

---

### 4. \( 2y = 3x \)

1. Solve for \( y \):  
   \( y = \frac{3x}{2} \)

This is already the equation for \( y \), and thus the inverse is the same.  
\( y = \frac{3x}{2} \)

---

### 5. \( i(x) = \frac{x - 5}{3} \)

1. Replace \( i(x) \) with \( y \):  
   \( y = \frac{x - 5}{3} \)
2. Swap \( x \) and \( y \):  
   \( x = \frac{y - 5}{3} \)
3. Solve for \( y \):  
   \( y - 5 = 3x \)  
   \( y = 3x + 5 \)

Thus, the inverse of \( i(x) \) is:  
\( i^{-1}(x) = 3x + 5 \)

---

Do you need further details or have any questions?

Here are 5 related questions:
1. How can we verify if a function and its inverse are correct?
2. What does it mean for a function to be one-to-one, and why is this important when finding inverses?
3. Can all functions have inverses? If not, which ones cannot?
4. How do graphical representations of a function and its inverse relate to each other?
5. How can we find the inverse of more complicated functions like logarithmic or exponential functions?

**Tip:** The inverse of a function can be thought of as "undoing" the effect of the original function. It reverses the inputs and outputs.

Find the inverse of the following functions: f(x) = x^3 - 15, g(x) = 2x/3, h(x) = x^2/5, 2y = 3x, i(x) = (x - 5)/3

Learn how to find the inverses of polynomial and linear functions with detailed solutions for equations like f(x) = x^3 - 15, g(x) = 2x/3, h(x) = x^2/5, and more. Understand key algebraic concepts and formulas. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation