Math Problem Statement
Solution
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:
Steps to find the inverse:
The inverse of a function is found by swapping the input-output pairs, i.e., if , then the inverse is .
Inverses of each function:
a) would be: Here, the function seems well-behaved, but for it to be a proper function, all the -values (which become the new -values) must be unique.
b) would be: This also appears to be a valid function with distinct -values (now -values).
c) would be: Again, this forms a function because the new -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 : May fail the invertibility test depending on the relation between , but as shown here, we can't confirm.
- Function : This is likely invertible since each input maps to a unique output.
- Function : Also seems invertible.
Conclusion:
Functions and are likely to be valid inverse functions, but for , 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:
- How do we verify if a function is one-to-one and invertible?
- What properties must a function satisfy to have an inverse?
- Can you explain the horizontal line test and its relation to invertibility?
- How do we graphically represent the inverse of a function?
- 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.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Inverse Functions
Relations
Formulas
If f = {(x1, y1), (x2, y2), ...}, then f^(-1) = {(y1, x1), (y2, x2), ...}
Theorems
Inverse function properties: One-to-one correspondence, uniqueness of output.
Suitable Grade Level
Grades 10-12