To determine the domain and range of the function [tex]\( y = \sqrt{x - 5} + 10 \)[/tex], let's break down the problem step-by-step.
### Domain:
1. Expression Inside the Square Root:
- The expression inside the square root is [tex]\( x - 5 \)[/tex].
- For the square root to be defined, the expression [tex]\( x - 5 \)[/tex] must be non-negative.
- Thus, we require:
[tex]\[
x - 5 \geq 0
\][/tex]
- Solving this inequality gives:
[tex]\[
x \geq 5
\][/tex]
- Therefore, the domain of the function is all [tex]\( x \)[/tex] such that [tex]\( x \geq 5 \)[/tex].
### Range:
1. Transformation by the Square Root:
- The square root function [tex]\(\sqrt{x - 5}\)[/tex] produces values starting from 0 (when [tex]\( x = 5 \)[/tex]) and increases as [tex]\( x \)[/tex] increases.
2. Shift by Addition:
- The function adds 10 to the square root component.
- Therefore, when [tex]\( x = 5 \)[/tex], [tex]\( y = \sqrt{5 - 5} + 10 = 0 + 10 = 10 \)[/tex].
- As [tex]\( x \)[/tex] increases, the value of [tex]\(\sqrt{x - 5}\)[/tex] increases, and consequently, [tex]\( y \)[/tex] also increases.
3. Range Based on Output:
- The minimum value of [tex]\( y \)[/tex] is 10 when [tex]\( x = 5 \)[/tex].
- Since there is no upper limit to the increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] can become arbitrarily large.
Combining these, the range of the function is all [tex]\( y \geq 10 \)[/tex].
### Summary:
- Domain: [tex]\( [5, \infty) \)[/tex]
- Range: [tex]\( [10, \infty) \)[/tex]
### Selection:
- Domain: [tex]\( x \geq 5 \)[/tex]
- Range: [tex]\( y \geq 10 \)[/tex]
