91Ó°ÊÓ

The absolute value function, noted as \(f(x) = |x|\), is a very interesting function given its unique behavior of mapping any real number to a non-negative output. This means that whether you input a positive or a negative number, the absolute value function will output the positive equivalent. For example, if you input \(x = -5\), the output will be \(f(x) = 5\). Similarly, if you input \(x = 5\), the output remains \(f(x) = 5\).

This behavior can be mathematically expressed as:

• If \(x > 0\), then \(|x| = x\).
• If \(x = 0\), then \(|x| = 0\).
• If \(x < 0\), then \(|x| = -x\).

This intrinsic property of the absolute value function explains why multiple inputs can lead to the same output. Understanding this behavior is key when analyzing whether the function is one-to-one.

When conducting a function analysis, we're essentially digging deep into the function's behavior and characteristics to understand its mapping between inputs and outputs. For the given function \(f(x) = |x|\), analyzing its behavior helps us identify if it is one-to-one or not.

The function maps both positive and negative values of \(x\) to the same non-negative value. For instance, both \(x = 4\) and \(x = -4\) result in the output \(|4| = 4\). This tells us that the function does not have a unique one-to-one mapping for each input to a singular output. Instead, it creates pairs of distinct inputs that lead to the same output. To determine if a function is one-to-one, we look for cases where \(f(a) = f(b)\), yet \(a eq b\). In the function \(f(x) = |x|\), there are clearly multiple such cases.

In the context of determining if a function is one-to-one, the concept of unique inputs and outputs is crucial. A one-to-one function must have a distinct input corresponding to each output. This means that no two different inputs should lead to the same output.

In the function \(f(x)=|x|\), this unique mapping doesn't exist. Consider the outputs for \(x = 3\) and \(x = -3\). Both yield the same result: \(f(3)=|3|=3\) and \(f(-3)=|-3|=3\). Here, we see that different inputs give us an identical output, reinforcing the fact that \(f(x)=|x|\) is not a one-to-one function. In mathematical terms, a function fails the horizontal line test if it maps distinct inputs to the same output, which is precisely what happens with \(f(x)=|x|\).

Understanding this ensures students can confidently distinguish between one-to-one functions and those that are not.