Answer :
To determine whether the equation [tex]\( x + 5 = y^2 \)[/tex] defines [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex], we need to consider the definition of a function. A function [tex]\( y = f(x) \)[/tex] assigns exactly one output [tex]\( y \)[/tex] for each input [tex]\( x \)[/tex].
Given the equation:
[tex]\[ x + 5 = y^2 \][/tex]
First, we solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ y^2 = x + 5 \][/tex]
[tex]\[ y = \pm \sqrt{x + 5} \][/tex]
From this, we see that for each value of [tex]\( x \)[/tex], there are two possible values of [tex]\( y \)[/tex]:
[tex]\[ y = \sqrt{x + 5} \][/tex]
and
[tex]\[ y = -\sqrt{x + 5} \][/tex]
This indicates that for any given [tex]\( x \)[/tex], there are two corresponding [tex]\( y \)[/tex] values (one positive and one negative). Thus, [tex]\( y \)[/tex] is not uniquely determined by [tex]\( x \)[/tex].
Since a function must assign exactly one output for each input, the equation [tex]\( x + 5 = y^2 \)[/tex] does not define [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex]. Therefore, the correct answer is:
No
Given the equation:
[tex]\[ x + 5 = y^2 \][/tex]
First, we solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ y^2 = x + 5 \][/tex]
[tex]\[ y = \pm \sqrt{x + 5} \][/tex]
From this, we see that for each value of [tex]\( x \)[/tex], there are two possible values of [tex]\( y \)[/tex]:
[tex]\[ y = \sqrt{x + 5} \][/tex]
and
[tex]\[ y = -\sqrt{x + 5} \][/tex]
This indicates that for any given [tex]\( x \)[/tex], there are two corresponding [tex]\( y \)[/tex] values (one positive and one negative). Thus, [tex]\( y \)[/tex] is not uniquely determined by [tex]\( x \)[/tex].
Since a function must assign exactly one output for each input, the equation [tex]\( x + 5 = y^2 \)[/tex] does not define [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex]. Therefore, the correct answer is:
No
