To determine the domain and range of the function [tex]\( y = \sqrt{x - 7} - 1 \)[/tex], let's analyze it step-by-step.
### Domain:
The domain of a function includes all the values for [tex]\( x \)[/tex] that make the function valid. In this function, we have a square root, which requires the expression inside it to be non-negative because we cannot take the square root of a negative number in the set of real numbers.
Let's set the expression inside the square root to be non-negative:
[tex]\[ x - 7 \geq 0 \][/tex]
Solving for [tex]\( x \)[/tex], we add 7 to both sides:
[tex]\[ x \geq 7 \][/tex]
Therefore, the domain of the function is:
[tex]\[ x \geq 7 \][/tex]
So, the domain is [tex]\( x \geq 7 \)[/tex].
### Range:
The range of a function includes all possible values that [tex]\( y \)[/tex] can take. In the function [tex]\( y = \sqrt{x - 7} - 1 \)[/tex], let's determine the minimum value of [tex]\( y \)[/tex].
1. The expression [tex]\( \sqrt{x - 7} \)[/tex] reaches its minimum value when [tex]\( x \)[/tex] is at its minimum value in the domain.
2. The smallest value [tex]\( x \)[/tex] can take is 7 (from the domain).
3. When [tex]\( x = 7 \)[/tex], the expression inside the square root is:
[tex]\[ \sqrt{7 - 7} = \sqrt{0} = 0 \][/tex]
4. Substituting this back into the function:
[tex]\[ y = \sqrt{0} - 1 = -1 \][/tex]
As [tex]\( x \)[/tex] increases beyond 7, [tex]\( \sqrt{x - 7} \)[/tex] becomes positive and thus [tex]\( y \)[/tex] becomes larger.
Thus, the minimum value of [tex]\( y \)[/tex] is [tex]\( -1 \)[/tex], and [tex]\( y \)[/tex] will take all values greater than or equal to [tex]\(-1\)[/tex].
Therefore, the range of the function is:
[tex]\[ y \geq -1 \][/tex]
So, the range is [tex]\( y \geq -1 \)[/tex].
### Final Answer:
- The domain is: [tex]\( x \geq 7 \)[/tex]
- The range is: [tex]\( y \geq -1 \)[/tex]
In the given context:
- The domain is [tex]\( x \geq \boxed{7} \)[/tex]
- The range is [tex]\( y \geq \boxed{-1} \)[/tex]
