Question

Which of the following could be an example of a function with a range (-infinite, a] and a domain [b, infinite) where a > 0 and b > 0

A.f(x)=-∛x+a -b
B.f(x)=∛(x+b)-a
C.f(x)=√x-a +b
D.f(x)=-√x-b +a

Answer 1

The correct answer is C.f(x) = √x - a + b.

Step-by-step explanation:

Step-by-step explanation:

The domain of the function is [b, ∞), which means that the input values (x) must be greater than or equal to b.

The range of the function is (-∞, a], which means that the output values (f(x)) must be less than or equal to a.

Let's check if option C meets these criteria:

For any x in the domain [b, ∞), the value √x - a + b will be greater than or equal to b - a + b = 2b - a, because √x ≥ 0 for all non-negative x. This means that the range of the function will be less than or equal to a if and only if 2b - a ≤ a, which is equivalent to 2b ≤ 2a, or b ≤ a. Since b > 0 and a > 0 by the problem statement, this condition is satisfied.

For any value of f(x) in the range (-∞, a], we can solve for x by rearranging the function: √x = f(x) - b + a. Since f(x) ≤ a, we have f(x) - b + a ≤ a - b + a = 2a - b, which means that √x ≤ 2a - b. Squaring both sides, we get x ≤ (2a - b)^2. Since 2a - b > 0 (because a > b), this condition is satisfied for any value of f(x) in the range (-∞, a].

Therefore, option C is a function that satisfies the given criteria.