f(x) = |x – 3| – 1
To understand this function, let’s break it down step by step
To understand this function, let’s break it down step by step.
The function is f(x) = |x – 3| – 1.
The vertical bars around (x – 3) indicate the absolute value function. It means that whatever is inside the vertical bars is taken as a positive value. In this case, we have (x – 3).
The next operation is subtraction, where we subtract 1 from the absolute value. So, we have |x – 3| – 1.
Let’s analyze the function for different values of x:
1. When x > 3:
In this case, (x – 3) is positive since x is greater than 3.
|x – 3| takes the positive value of (x – 3).
So, f(x) = (x – 3) – 1 = x – 4.
2. When x < 3:
In this case, (x - 3) is negative since x is less than 3.
|x - 3| takes the positive value of -(x - 3) = 3 - x.
So, f(x) = -(x - 3) - 1 = 4 - x.
3. When x = 3:
In this case, (x - 3) equals 0.
|x - 3| takes the positive value of 0, which is 0.
So, f(x) = 0 - 1 = -1.
To summarize:
- For x > 3, f(x) = x – 4.
– For x < 3, f(x) = 4 - x.
- For x = 3, f(x) = -1.
This means that the graph of the function f(x) = |x - 3| - 1 will consist of two segments: one with a positive slope (x - 4) for x greater than 3, and one with a negative slope (4 - x) for x less than 3. At x = 3, there will be a point at (-1, 3) where the two segments meet.
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