Understanding the Range of a Function: Example with the Function f(x) = -2 |x+1|

What is the range of the function?f(x) = -2 Ix+1I

B.

The range of a function refers to the set of all the possible output values (y-values) that the function could produce for any given input.

In the given function, f(x) = -2 |x+1|, there are two cases to consider based on the value of the input x:

1. If x+1 is positive, then |x+1| = x+1, and the function simplifies to f(x) = -2(x+1) = -2x-2.
2. If x+1 is negative, then |x+1| = -(x+1), and the function simplifies to f(x) = -2(-(x+1)) = 2x+2.

Therefore, for any given input x, the function f(x) can produce output values that fall under two categories:
– If x ≤ -1, then the output will be f(x) = 2x+2, which can take any value from (-∞, 0].
– If x ≥ -1, then the output will be f(x) = -2x-2, which can take any value from [-4, ∞).

Thus, the range of the given function f(x) is (-∞, 0] ∪ [-4, ∞).

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