Let g(x)= {1 if x≥0
−1 if x<0.


a. Write a formula for |g(x)|.

Verified step by step guidance

1. Step 1: Understand the function g(x). It is a piecewise function defined as g(x) = 1 if x \(\geq\) 0 and g(x) = -1 if x < 0.
2. Step 2: Recall the definition of the absolute value function. For any real number y, |y| is y if y \(\geq\) 0 and -y if y < 0.
3. Step 3: Apply the absolute value definition to g(x). Since g(x) is either 1 or -1, consider each case separately.
4. Step 4: For x \(\geq\) 0, g(x) = 1, and |g(x)| = |1| = 1.
5. Step 5: For x < 0, g(x) = -1, and |g(x)| = |-1| = 1.

Key Concepts

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, g(x) is defined as 1 for x ≥ 0 and -1 for x < 0. Understanding how to interpret and manipulate piecewise functions is essential for determining their properties and transformations.

Absolute Value Function

The absolute value function, denoted as |x|, represents the distance of x from zero on the number line, always yielding a non-negative result. For the function g(x), the absolute value |g(x)| will convert any negative outputs to positive, which is crucial for writing the formula for |g(x)|.

Function Composition

Function composition involves applying one function to the results of another. In this context, to find |g(x)|, we need to apply the absolute value operation to the outputs of the piecewise function g(x). This concept is fundamental in understanding how to combine different mathematical operations effectively.