Based on the given function, the range of h(x) is f(x) ≤ 5. The correct answer is B) f(x) ≤ 5.
How to find the range
The function h(x) is defined as shown:
h(x) = x + 2, when x < 3
h(x) = -x + 8, when x ≥ 3
The question asks about the range of h(x). This means determining the set of possible output values (f(x)) for the function h(x).
For when x < 3:
As x decreases towards negative infinity, h(x) also decreases towards negative infinity.
The smallest value for this domain is h(2) = 2 + 2 = 4.
For when x ≥ 3:
As x increases towards positive infinity, h(x) decreases towards negative infinity.
The largest value for this domain is h(3) = -3 + 8 = 5.
Hence, the range of h(x) is f(x) ≤ 5 or f(x) ≥ -∞.
Among the options provided:
A) -∞ < f(x) < ∞: This represents the entire real number line, but it's not specific to the range of h(x).
B) f(x) ≤ 5: This correctly represents the range of h(x).
C) f(x) ≥ 5: This does not cover values less than 5, so it's not the complete range of h(x).
D) f(x) ≥ 3: This only covers values greater than or equal to 3, which is not the entire range of h(x).
Therefore, the correct answer among the given options is B) f(x) ≤ 5.
The function h(x) is defined as shown.
What is the range of h(x)?
A) –∞ < f(x) < ∞
B) f(x) ≤ 5
C) f(x) ≥ 5
D) f(x) ≥ 3
h(x)= { x + 2, x < 3
{ -x + 8. x ≥ 3