Which of the following could be an example of a function with a range $(-\infty, a]$ and a domain $[b, \infty)$ where $a>0$ and $b>0$?
A. $f(x)=-\sqrt{x-b}+a$
B. $f(x)=\sqrt[3]{(x+b)}-a$
C. $f(x)=\sqrt{x-a}+b$
D. $f(x)=-\sqrt[3]{x+a}-b$
Answer (1)
Analyze the domain and range of each function.
Compare the domain and range of each function with the given domain [ b , ∞ ) and range ( − ∞ , a ] .
Function A, f ( x ) = − x − b + a , has the domain [ b , ∞ ) and range ( − ∞ , a ] .
Therefore, the correct answer is A .
Explanation
Understanding the Problem We are given the range ( − ∞ , a ] and the domain [ b , ∞ ) where 0"> a > 0 and 0"> b > 0 . We need to find the function that satisfies these conditions.
Analyzing Each Option Let's analyze each option:
A. f ( x ) = − x − b + a
Domain: x − b ≥ 0 ⟹ x ≥ b . So the domain is [ b , ∞ ) .
Range: Since x − b ≥ 0 , then − x − b ≤ 0 , so − x − b + a ≤ a . Thus the range is ( − ∞ , a ] .
B. f ( x ) = 3 ( x + b ) − a
Domain: all real numbers.
Range: all real numbers.
C. f ( x ) = x − a + b
Domain: x − a ≥ 0 ⟹ x ≥ a . So the domain is [ a , ∞ ) .
Range: Since x − a ≥ 0 , then x − a + b ≥ b . Thus the range is [ b , ∞ ) .
D. f ( x ) = − 3 x + a − b
Domain: all real numbers.
Range: all real numbers.
Finding the Matching Function Comparing the domain and range of each function with the given domain [ b , ∞ ) and range ( − ∞ , a ] , we see that function A matches the given conditions.
Conclusion Therefore, the correct answer is A. f ( x ) = − x − b + a .
Examples
Understanding the domain and range of functions is crucial in various real-world applications. For instance, when modeling the height of a projectile over time, the domain represents the time interval during which the projectile is in the air, and the range represents the possible heights it can reach. Similarly, in economics, the domain of a cost function might represent the number of units produced, while the range represents the total cost of production. By analyzing the domain and range, we can gain valuable insights into the behavior and limitations of the model.
