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A one-to-one function is a special type of function that maps each input to a unique output. No two different inputs in the domain should result in the same output. To determine whether a function is one-to-one, you need to examine how the outputs behave as the inputs change. If a function is monotonic, it is often one-to-one. This means the function should continuously increase or decrease across its entire domain, without any plateaus or turning points.

Checking the function from our exercise, \( f(x) = x + |x| \), we analyzed the input values by splitting it into positive and negative domains. For positive values, the function behaves as \( f(x) = 2x \), giving unique outputs for unique inputs. But, for negative values, it always maps to \( f(x) = 0 \). Therefore, it is not considered one-to-one across its entire domain, as multiple inputs (all negative \( x \)) provide the same output.

The horizontal line test is a useful tool to determine if a function is one-to-one. By visually inspecting the graph of a function, you can establish if each output corresponds to only one input. This test involves drawing horizontal lines (parallel to the x-axis) across the function's graph.

If any horizontal line intersects the graph more than once, then the function is not one-to-one. That means the same output value corresponds to multiple input values. For the given function, \( f(x) = x + |x| \), the graph reveals that all outputs are unique for positive \( x \) values. However, all negative \( x \) converge to \( f(x) = 0 \), causing any horizontal line drawn at \( y = 0 \) to intersect numerous times. Hence, failing the horizontal line test shows the function does not have an inverse.

The domain of a function refers to all possible input values (usually \( x \)-values), while the range encompasses all possible output values (or \( y \)-values) that the function can produce.

For \( f(x) = x + |x| \), the domain consists of all real numbers. But the function behaves differently based on the value of \( x \) being positive or negative. When \( x \geq 0 \), the function behaves as a linear function, producing outputs from 0 to infinity, thus forming the range \([0, \infty)\).

• If \( x < 0 \): \( f(x) = 0 \), a constant output.
• If \( x \geq 0 \): \( f(x) = 2x \), a line starting at \( (0,0) \)

Understanding these aspects helps determine the behavior and characteristics of a function, especially when looking for inverses.

An absolute value function is represented by \( |x| \) and describes the distance of the number \( x \) from zero on the number line, effectively making all output values non-negative. It operates under two conditions: it returns \( x \) when \( x \) is zero or positive, and returns \(-x\) when \( x \) is negative.

In the function \( f(x) = x + |x| \), the absolute value component splits the function's behavior into two distinct parts:

• For \( x \geq 0 \), \( |x| = x \), so the function simplifies to \( 2x \).
• For \( x < 0 \), \( |x| = -x \), so the function evaluates to \( 0 \).

This characteristic of changing behavior with sign changes makes absolute value functions integral to discussing the general behavior of \( f(x) = x + |x| \) and illustrates why it cannot have an inverse over its full domain.