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

Verified step by step guidance

1. Recall the definition of the absolute value function: for any real number \(x\), \(|x|\) represents the distance of \(x\) from zero on the number line, which is always non-negative.
2. Write down the two functions explicitly: \(f(x) = |x|\) and \(g(x) = |-x|\).
3. Use the property of absolute value that states \(|a| = |-a|\) for any real number \(a\). Applying this to \(g(x)\), we get \(g(x) = |-x| = |x|\).
4. Conclude that since \(g(x)\) simplifies to \(|x|\), the graphs of \(f(x)\) and \(g(x)\) are identical.
5. Therefore, the relationship between the graphs of \(f(x)\) and \(g(x)\) is that they coincide exactly; they are the same graph.

Key Concepts

Absolute Value Function

The absolute value function, denoted |x|, outputs the non-negative value of x regardless of its sign. It creates a V-shaped graph symmetric about the y-axis, reflecting all negative inputs as positive outputs.

Function Transformation and Symmetry

Understanding how transformations affect graphs is key. Replacing x with -x reflects the graph across the y-axis. Since |x| is symmetric about the y-axis, this transformation does not change the graph's shape or position.

Graph Comparison

Comparing graphs involves analyzing their shapes, positions, and symmetries. Since ƒ(x) = |x| and g(x) = |-x| produce identical outputs for all x, their graphs coincide, illustrating that g(x) is essentially the same function as ƒ(x).