show that f(x)=|x-1| is not a one-to-one function

To demonstrate that the function \( f(x) = |x - 1| \) is not a one-to-one function, we need to show that there exist at least two distinct values of \( x \) for which the function yields the same output. 

### Step 1: Define the Function
The function given is:
\[
f(x) = |x - 1|
\]
This function represents the absolute value of \( x - 1 \).

### 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 \( x_1 \) and \( x_2 \) such that \( f(x_1) = f(x_2) \) but \( x_1 \neq x_2 \).

Let's consider \( x_1 = 0 \) and \( x_2 = 2 \).

- For \( x_1 = 0 \):
\[
f(0) = |0 - 1| = |-1| = 1
\]

- For \( x_2 = 2 \):
\[
f(2) = |2 - 1| = |1| = 1
\]

### Step 3: Conclusion
We have \( f(0) = 1 \) and \( f(2) = 1 \), but \( 0 \neq 2 \). This shows that \( f(x) \) gives the same output (1) for two different inputs (0 and 2). 

Therefore, \( f(x) = |x - 1| \) is **not** a one-to-one function because it does not satisfy the requirement that each output corresponds to exactly one unique input.

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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:

1. What is the formal definition of a one-to-one (injective) function?
2. How can you determine if a function is one-to-one using a graph?
3. What is the difference between one-to-one and onto functions?
4. Can you give an example of a one-to-one function and explain why it is so?
5. 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.

Explore how to demonstrate that the function f(x) = |x - 1| is not one-to-one through step-by-step analysis. Learn about absolute value functions and their behavior, with examples showing how multiple inputs can yield the same output.

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