To identify the parent function of [tex]\( f(x) = 6 + \sqrt{x + 2} - 5 \)[/tex], let's break down the expression step by step.
The given function is:
[tex]\[ f(x) = 6 + \sqrt{x + 2} - 5 \][/tex]
First, simplify the expression inside the function:
[tex]\[ f(x) = 6 - 5 + \sqrt{x + 2} \][/tex]
[tex]\[ f(x) = 1 + \sqrt{x + 2} \][/tex]
Next, focus on the transformation applied to [tex]\( x \)[/tex] within the square root. The term [tex]\( x + 2 \)[/tex] indicates a horizontal shift of the base parent function [tex]\( y = \sqrt{x} \)[/tex]. This horizontal shift moves the graph to the left by 2 units.
If we ignore the constant term (+1) and the horizontal shift momentarily, the core of the function which shows the transformation of [tex]\( x \)[/tex] is [tex]\( \sqrt{x} \)[/tex].
Therefore, the parent function of [tex]\( f(x) = 6 + \sqrt{x + 2} - 5 \)[/tex] without any shifts or additional constants is:
[tex]\[ y = \sqrt{x} \][/tex]
Among the given options, the correct parent function is:
[tex]\[ y = \sqrt{x} \][/tex]
