What is the relationship between the graphs of ƒ(x)=|x| and g(x)=|-x|?

Key Concepts

Absolute Value Function

The absolute value function, denoted as |x|, outputs the non-negative value of x regardless of its sign. This means that for any real number x, |x| is equal to x if x is positive or zero, and -x if x is negative. The graph of ƒ(x) = |x| is a V-shaped curve that opens upwards, with its vertex at the origin (0,0).

Reflection Across the Y-Axis

The function g(x) = |-x| represents a reflection of the function ƒ(x) = |x| across the y-axis. This is because replacing x with -x in the function does not change the output of the absolute value, as both positive and negative inputs yield the same result. Thus, the graphs of ƒ(x) and g(x) are identical, demonstrating symmetry about the y-axis.

Graphical Symmetry

Graphical symmetry refers to the property of a graph where one side mirrors the other across a specific line, such as the y-axis. In this case, both ƒ(x) = |x| and g(x) = |-x| exhibit this symmetry, as their graphs are the same. Understanding this concept helps in analyzing how transformations affect the shape and position of graphs in the coordinate plane.