Question

Suppose that f (x) = 0 when x ≤ 0,

f (x) = x when 0 < x ≤ 1,
f (x) = 2 − x when 1 < x < 2, and
f (x) = 0 when x ≥ 2.
compute f (x) in each of the ranges x ≤ 0, 0 < x ≤ 1, 1 < x < 2, and x ≥ 2.

Answer 1

In the given function (f(x)):

1. For
`\(x \leq 0\)`, the function is defined as 0.

2. For
`\(0 < x \leq 1\)`, the function is equal to the input value, (f(x) = x).

3. For
`\(1 < x < 2\)`, the function is (f(x) = 2 - x).

4. For
`\(x \geq 2\)`, the function becomes 0.

Explanation:

In the given piecewise function (f(x)), different rules govern the behavior of the function in distinct intervals. Firstly, for
`\(x \leq 0\)`, the function is defined as (f(x) = 0). This means that for any (x) less than or equal to zero, the output of the function is zero. Moving to the interval
`\(0 < x \leq 1\)`, the function takes the form
`\(f(x) = x\)`, signifying that within this range, the output is equivalent to the input value.

When (1 < x < 2), the function becomes (f(x) = 2 - x), indicating a linear decrease as (x) increases within this interval. Lastly, for
`\(x \geq 2\)`, the function reverts to (f(x) = 0), rendering any input equal to or greater than 2 to result in an output of zero.

These rules create a segmented function, each part uniquely determining the output based on the input's position within specific intervals. Such piecewise functions are valuable in mathematical modeling, allowing the representation of diverse behaviors within a single expression. They find applications in various fields, including physics, economics, and engineering, where phenomena exhibit distinct patterns or rules under different conditions.