Slope functions Determine the slope function S (x) for the following functions
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Verified step by step guidance

1. Step 1: Understand the function \( f(x) = |x| \). The absolute value function is defined as \( f(x) = x \) if \( x \geq 0 \) and \( f(x) = -x \) if \( x < 0 \).
2. Step 2: Consider the piecewise nature of the function. For \( x > 0 \), the function is \( f(x) = x \), and for \( x < 0 \), the function is \( f(x) = -x \).
3. Step 3: Differentiate each piece separately. For \( x > 0 \), the derivative \( f'(x) = 1 \). For \( x < 0 \), the derivative \( f'(x) = -1 \).
4. Step 4: Consider the point \( x = 0 \). The function \( f(x) = |x| \) is not differentiable at \( x = 0 \) because the left-hand derivative and the right-hand derivative are not equal.
5. Step 5: Combine the results to form the slope function \( S(x) \). Thus, \( S(x) = 1 \) for \( x > 0 \), \( S(x) = -1 \) for \( x < 0 \), and \( S(x) \) is undefined at \( x = 0 \).

Key Concepts

Derivative

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is often interpreted as the slope of the tangent line to the function's graph at that point. For the function f(x) = |x|, the derivative will vary depending on whether x is positive, negative, or zero, leading to different slope values.

Absolute Value Function

The absolute value function, denoted as |x|, outputs the non-negative value of x regardless of its sign. This function is piecewise defined: it equals x when x is positive and -x when x is negative. Understanding this behavior is crucial for determining the slope function, as it affects the derivative's definition across different intervals.

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. For the absolute value function, f(x) = |x| can be expressed as f(x) = x for x ≥ 0 and f(x) = -x for x < 0. Recognizing how to handle piecewise functions is essential for calculating derivatives, as it requires analyzing each segment separately to find the overall slope function.