https://file.aiquickdraw.com/mathbot/file/1728389922562ult1idr6.jpg

The question asks to find the inverse of the given functions and determine which of them represent actual inverse functions. There are three relations defined by their ordered pairs:

1. \( f = \{(a,b), (b,c), (c,d), (d,e)\} \)
2. \( g = \{(x,-2), (0,2), (1,-1), (2,-3)\} \)
3. \( h = \{(-1,6), (0,-1), (2,2), (6,0)\} \)

### Steps to find the inverse:
The inverse of a function \( f(x) \) is found by swapping the input-output pairs, i.e., if \( f = \{(x_1, y_1), (x_2, y_2), \dots \} \), then the inverse \( f^{-1} \) is \( \{(y_1, x_1), (y_2, x_2), \dots \} \).

### Inverses of each function:
a) \( f^{-1} \) would be:
\[
f^{-1} = \{(b,a), (c,b), (d,c), (e,d)\}
\]
Here, the function seems well-behaved, but for it to be a proper function, all the \( y \)-values (which become the new \( x \)-values) must be unique.

b) \( g^{-1} \) would be:
\[
g^{-1} = \{(-2,x), (2,0), (-1,1), (-3,2)\}
\]
This also appears to be a valid function with distinct \( y \)-values (now \( x \)-values).

c) \( h^{-1} \) would be:
\[
h^{-1} = \{(6,-1), (-1,0), (2,2), (0,6)\}
\]
Again, this forms a function because the new \( x \)-values are unique.

### Determining which is a valid inverse function:
- A function is invertible if no two different inputs produce the same output (i.e., it passes the horizontal line test for invertibility). Checking this:
  - **Function \( f \)**: May fail the invertibility test depending on the relation between \( a, b, c, d, e \), but as shown here, we can't confirm.
  - **Function \( g \)**: This is likely invertible since each input maps to a unique output.
  - **Function \( h \)**: Also seems invertible.

### Conclusion:
Functions \( g \) and \( h \) are likely to be valid inverse functions, but for \( f \), additional clarification about the values would be needed.

---

Would you like more details or have any questions? Here are five related questions to expand the topic:

1. How do we verify if a function is one-to-one and invertible?
2. What properties must a function satisfy to have an inverse?
3. Can you explain the horizontal line test and its relation to invertibility?
4. How do we graphically represent the inverse of a function?
5. Are there cases where functions have partial inverses?

**Tip**: Always check if a function is one-to-one before finding its inverse, as only one-to-one functions have true inverses.

Tentukan invers dari fungsi berikut dan selidiki mana yang merupakan fungsi invers: a) f = {(a,b), (b,c), (c,d), (d,e)}, b) g = {(x,-2), (0,2), (1,-1), (2,-3)}, c) h = {(-1,6), (0,-1), (2,2), (6,0)}

Learn how to find inverse functions from given relations and verify if they form valid inverse functions. This example includes step-by-step analysis of three functions, demonstrating key algebraic principles such as function invertibility and uniqueness. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation