To find the domain and range of the function [tex]\( f(x) = \sqrt{x - 7} + 9 \)[/tex], we must determine the values of [tex]\( x \)[/tex] for which the function is defined and the range of output values [tex]\( y \)[/tex] that it can take.
### Domain
Firstly, consider the expression inside the square root: [tex]\( x - 7 \)[/tex]. For the square root to be defined, the expression inside must be non-negative. Therefore:
[tex]\[ x - 7 \geq 0 \][/tex]
Solving this inequality:
[tex]\[ x \geq 7 \][/tex]
Thus, the domain of the function is:
[tex]\[ x \geq 7 \][/tex]
### Range
Now, consider the output values [tex]\( y \)[/tex]. The function [tex]\( f(x) = \sqrt{x - 7} + 9 \)[/tex] starts with the square root, which has a minimum value of 0 when [tex]\( x = 7 \)[/tex].
When [tex]\( x = 7 \)[/tex]:
[tex]\[ f(7) = \sqrt{7 - 7} + 9 = 0 + 9 = 9 \][/tex]
As [tex]\( x \)[/tex] increases beyond 7, the value of [tex]\( \sqrt{x - 7} \)[/tex] increases without bound, which means [tex]\( y \)[/tex] will also increase without bound.
Therefore, the minimum value of [tex]\( y \)[/tex] is 9, and it increases infinitely.
So, the range of the function is:
[tex]\[ y \geq 9 \][/tex]
### Conclusion
Based on our analysis, the correct domain and range of the function [tex]\( f(x) = \sqrt{x - 7} + 9 \)[/tex] are:
- Domain: [tex]\( x \geq 7 \)[/tex]
- Range: [tex]\( y \geq 9 \)[/tex]
Hence, the correct choice is:
[tex]\[ \text{Domain: } x \geq 7, \text{ Range: } y \geq 9 \][/tex]
