Question

The range of the function \( f(x) = 8 + \sqrt{x - 5} \) is:

• A. \( (-\infty, 5] \)
• B. \( [5, \infty) \)
• C. \( (-\infty, 5] \cup [8, \infty) \)
• D. \( [5, 8] \)
• E. \( [8, \infty) \)

Hint:

For functions involving square roots, ensure that the expression under the square root is non-negative. This will help in determining the domain and range.

Answer 1

Answer: (E)

For \( f(x) = 8 + \sqrt{x - 5} \), the square root function is defined only when \( x - 5 \geq 0 \), 
so \( x \geq 5 \).
Therefore, the smallest value of \( f(x) \) occurs when \( x = 5 \), giving \( f(5) = 8 \). As \( x \) increases, \( \sqrt{x - 5} \) increases, so \( f(x) \) increases. 
Thus, the range of the function is \( [8, \infty) \). 
Thus, the correct answer is (E).