To determine the range of the function [tex]\( f(x) = \sqrt{x - 8} + 6 \)[/tex], we'll follow a step-by-step process:
1. Domain Determination:
- First, recognize that inside a square root function, the expression must be non-negative. That is, we need [tex]\( x - 8 \geq 0 \)[/tex].
- Solving for [tex]\( x \)[/tex], we get [tex]\( x \geq 8 \)[/tex].
2. Behavior of the Function:
- Let's examine [tex]\( f(x) \)[/tex] when [tex]\( x = 8 \)[/tex]:
[tex]\[
f(8) = \sqrt{8 - 8} + 6 = \sqrt{0} + 6 = 6
\][/tex]
- As [tex]\( x \)[/tex] increases beyond 8, [tex]\( x - 8 \)[/tex] becomes positive and grows larger.
- The square root function [tex]\( \sqrt{x - 8} \)[/tex] will yield non-negative values because the square root of a non-negative number is also non-negative.
3. Range Calculation:
- The smallest value of the square root function [tex]\( \sqrt{x - 8} \)[/tex] when [tex]\( x = 8 \)[/tex] is 0, which makes the minimum value of [tex]\( f(x) \)[/tex]: [tex]\( 0 + 6 = 6 \)[/tex].
- As [tex]\( x \)[/tex] increases, [tex]\( \sqrt{x - 8} \)[/tex] can take any non-negative value, becoming larger with increasing [tex]\( x \)[/tex]. Therefore, [tex]\( f(x) \)[/tex] can take any value starting from 6 and increasing without bound.
- In other words, [tex]\( f(x) \geq 6 \)[/tex].
So, the range of the function [tex]\( f(x) = \sqrt{x - 8} + 6 \)[/tex] is:
[tex]\[ f(x) \geq 6 \][/tex]
Among the given options, the correct answer is:
[tex]\[ f(x) \geq 6 \][/tex]
This corresponds to the third option:
[tex]\[ f(x) \geq 6 \][/tex]
