Math Problem Statement
show that f(x)=|x-1| is not a one-to-one function
Solution
To demonstrate that the function is not a one-to-one function, we need to show that there exist at least two distinct values of for which the function yields the same output.
Step 1: Define the Function
The function given is: This function represents the absolute value of .
Step 2: Find Distinct Values with the Same Output
To check if the function is not one-to-one, we need to find two different values and such that but .
Let's consider and .
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For :
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For :
Step 3: Conclusion
We have and , but . This shows that gives the same output (1) for two different inputs (0 and 2).
Therefore, is not a one-to-one function because it does not satisfy the requirement that each output corresponds to exactly one unique input.
Would you like to dive deeper into any part of this explanation or explore related topics? Here are five related questions you might find interesting:
- What is the formal definition of a one-to-one (injective) function?
- How can you determine if a function is one-to-one using a graph?
- What is the difference between one-to-one and onto functions?
- Can you give an example of a one-to-one function and explain why it is so?
- How would you prove that a quadratic function is not one-to-one?
Tip: To check if a function is one-to-one, you can also use the horizontal line test on its graph. If any horizontal line intersects the graph more than once, the function is not one-to-one.
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Math Problem Analysis
Mathematical Concepts
Functions
Absolute Value
One-to-One Functions
Formulas
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Theorems
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Suitable Grade Level
High School