Math Problem Statement

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

Solution

To demonstrate that the function f(x)=x1f(x) = |x - 1| is not a one-to-one function, we need to show that there exist at least two distinct values of xx for which the function yields the same output.

Step 1: Define the Function

The function given is: f(x)=x1f(x) = |x - 1| This function represents the absolute value of x1x - 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 x1x_1 and x2x_2 such that f(x1)=f(x2)f(x_1) = f(x_2) but x1x2x_1 \neq x_2.

Let's consider x1=0x_1 = 0 and x2=2x_2 = 2.

  • For x1=0x_1 = 0: f(0)=01=1=1f(0) = |0 - 1| = |-1| = 1

  • For x2=2x_2 = 2: f(2)=21=1=1f(2) = |2 - 1| = |1| = 1

Step 3: Conclusion

We have f(0)=1f(0) = 1 and f(2)=1f(2) = 1, but 020 \neq 2. This shows that f(x)f(x) gives the same output (1) for two different inputs (0 and 2).

Therefore, f(x)=x1f(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.


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.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Functions
Absolute Value
One-to-One Functions

Formulas

-

Theorems

-

Suitable Grade Level

High School