Analyzing the Function f(x) = |x – 3| – 1: Definition, Cases, and Graph Analysis

f(x) = |x – 3| – 1

To understand and analyze this function, let’s break it down step by step

To understand and analyze this function, let’s break it down step by step.

The function you provided is f(x) = |x – 3| – 1.

The symbol |x – 3| indicates the absolute value of the expression (x – 3).

The absolute value function is defined as follows:
– If the expression within the absolute value bars is positive, the absolute value is equal to the expression.
– If the expression within the absolute value bars is negative, the absolute value is equal to the negation of the expression (making it positive).

Now, let’s consider the function in different cases:

Case 1: When (x – 3) ≥ 0
In this case, the expression (x – 3) is non-negative or positive. Therefore, the absolute value |x – 3| will be equal to (x – 3) itself.
Hence, f(x) = (x – 3) – 1 = x – 4.

Case 2: When (x – 3) < 0 In this case, the expression (x - 3) is negative. Therefore, the absolute value |x - 3| will be equal to -(x - 3) (negating the negative value to make it positive). Hence, f(x) = -(x - 3) - 1 = -x + 3 - 1 = -x + 2. To summarize: - For x ≥ 3, f(x) = x - 4 - For x < 3, f(x) = -x + 2 Now, let's examine the graph of this function: The graph of f(x) = |x - 3| - 1 consists of two linear segments. The first segment corresponds to x < 3, where f(x) = -x + 2. This line has a negative slope of -1, intersecting the y-axis at (0, 2). The second segment corresponds to x ≥ 3, where f(x) = x - 4. This line has a positive slope of 1, intersecting the y-axis at (0, -4). At x = 3, the function experiences a discontinuity (also known as a corner) because the two segments meet. However, the value of the function is the same from both directions. In summary, the function f(x) = |x - 3| - 1 has a piecewise definition: - For x < 3, the graph is a line with a negative slope: f(x) = -x + 2. - For x ≥ 3, the graph is a line with a positive slope: f(x) = x - 4.

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