To determine the range of the function [tex]\( y = \sqrt{x + 7} + 5 \)[/tex], we need to analyze how the function behaves with respect to [tex]\( x \)[/tex].
1. Expression under the square root:
Since the function involves a square root, the expression inside the square root must be non-negative. Hence, for [tex]\( y = \sqrt{x + 7} + 5 \)[/tex], the term [tex]\( x + 7 \)[/tex] must be greater than or equal to 0.
[tex]\[
x + 7 \geq 0
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x \geq -7
\][/tex]
2. Behavior of the function:
Let's examine how the function behaves at the boundary point [tex]\( x = -7 \)[/tex]:
[tex]\[
y = \sqrt{-7 + 7} + 5 = \sqrt{0} + 5 = 0 + 5 = 5
\][/tex]
Therefore, when [tex]\( x = -7 \)[/tex], [tex]\( y \)[/tex] is 5.
3. Increasing function:
For [tex]\( x > -7 \)[/tex], [tex]\( x + 7 \)[/tex] is positive, and hence [tex]\( \sqrt{x + 7} \)[/tex] will yield values greater than 0. As a result, [tex]\( y \)[/tex] will be greater than 5 since it is the sum of a positive term ([tex]\( \sqrt{x + 7} \)[/tex]) and 5.
Since the minimum value of [tex]\( y \)[/tex] is 5 when [tex]\( x = -7 \)[/tex] and [tex]\( y \)[/tex] increases without any upper bound as [tex]\( x \)[/tex] increases, the range of [tex]\( y \)[/tex] is:
[tex]\[
y \geq 5
\][/tex]
Thus, the range of the function [tex]\( y = \sqrt{x + 7} + 5 \)[/tex] is:
[tex]\[
\boxed{y \geq 5}
\][/tex]
