What is its Domain and Range.

Question

The equation is f(x)=-4|x- 2|+ 3.

Answer 1

Finding the domain:

For x ≥ 2, we have f(x) = -4|x - 2|+ 3 = -4(x - 2) + 3 = -4x + 11 which clearly gives a well-defined output.

For x ≤ 2, we have f(x) = -4|x - 2| + 3 = -4(2 - x) + 3 = 4x - 5 which also clearly gives a well-defined output.

Taking union of x ≥ 2 and x ≤ 2 gives all real numbers.

Therefore, the domain of f is all real numbers (-∞ < x < ∞).

Finding the Range:

If x ≥ 2,

-4x ≤ -8, (Multiplying both sides of the inequality by -4.)

-4x + 11 ≤ 3, (Adding 11 on both sides of the inequality.)

f(x) ≤ 3 (Since f(x) = -4|x - 2| + 3 = -4x + 11 when x ≥ 2.)

If x ≤ 2,

4x ≤ 8, (Multiplying both sides of the inequality by 4.)

4x - 5 ≤ 3, (Subtracting 5 from both sides of the inequality.)

f(x) ≤ 3. (Since f(x) = -4|x - 2| + 3 = -4x + 11 when x ≤ 2.)

Therefore, if we input any value of x into our function f over x ≥ 2 or x ≤ 2, which is the domain of all real numbers, we'll have the range -∞ < f(x) ≤ 3.

Answer 2

Domain: (-∞, ∞);

Range: (- ∞, 3]