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When analyzing the function graphically to determine whether it is one-to-one, we look at how the graph behaves visually. For the function \( f(x) = |x-3| \), it forms a V-shaped graph, where the vertex of the V is located at the point \((3, 0)\).
The graph is symmetric along the vertical line called the axis of symmetry, which in this case is the line \( x = 3 \).
The symmetry of the graph implies that for any point to the left of the vertical line, there is a corresponding point to the right, both having the same \(y\)-value. For example, the points \((2, 1)\) and \((4, 1)\), both feature the same height on the graph showing that they have the same output although their inputs differ.
This mirroring effect indicates that the graph is not one-to-one, since different \(x\)-values lead to identical \(y\)-values, violating the one-to-one function condition.

The algebraic method involves analyzing the equations to check conditions for a one-to-one function. If we set \( f(x) = |x-3| \) equal to another instance of \( f \), as \( f(a) = f(b) \), it results in the modulus equation \( |a - 3| = |b - 3| \).
This equation implies two situations: either \( a - 3 = b - 3 \) or \( a - 3 = -(b - 3) \).
In the first scenario, \( a = b \), which supports a one-to-one relationship.
However, the second scenario gives \( a = 6-b \), indicating \( a \) and \( b \) can have different values but yield the same result when substituted back into the function.
This breakdown shows algebraically that \( f(x) = |x - 3| \) is not one-to-one because it is possible for different inputs \( a \) and \( b \) to result in the same output, confirming multiple inputs for one output.

The absolute value function is a cornerstone in understanding how functions can behave differently compared to more straightforward linear functions. By definition, the absolute value \( |x| \) measures the distance of \( x \) from zero on the number line, regardless of direction.
When modified into a function like \( f(x) = |x-3| \), it essentially shifts the typical absolute value function horizontally by 3 units to the right.
This change creates the described V shape with a vertex at \( (3, 0) \).
The key attribute of an absolute value function is its tendency to create symmetrical outputs over its vertex, such as inputs like \( x=2 \) and \( x=4 \), both generating the same output under the function's definition.
Without this symmetry, a function might display a linear or parabolic characteristic where one-to-one conditions are more favorable. Understanding these properties helps clarify why, in this specific case, the function \( f(x)=|x-3| \) does not qualify as a one-to-one function.