To determine which of the given equations represents a linear function, let's analyze each equation individually.
1. [tex]\( y = x^2 + 3 \)[/tex]
In this equation, the term [tex]\( x^2 \)[/tex] indicates a quadratic relationship. A quadratic equation generally takes the form [tex]\( y = ax^2 + bx + c \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants and [tex]\( a \neq 0 \)[/tex]. Quadratic equations are not linear because they involve the square of the variable [tex]\( x \)[/tex].
2. [tex]\( x = y^2 - 2 \)[/tex]
Here, the term [tex]\( y^2 \)[/tex] indicates a quadratic relationship in [tex]\( y \)[/tex]. Additionally, we can rewrite it as [tex]\( y^2 = x + 2 \)[/tex], showing that [tex]\( y \)[/tex] is a quadratic function of [tex]\( x \)[/tex]. This is also not a linear relationship, as it involves the square of the variable [tex]\( y \)[/tex].
3. [tex]\( x = 1 \)[/tex]
This equation describes a vertical line where [tex]\( x \)[/tex] is always equal to 1, regardless of the value of [tex]\( y \)[/tex]. While it is a constant function in terms of [tex]\( x \)[/tex], it does not fit the general form of a linear function [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] and [tex]\( b \)[/tex] are constants.
4. [tex]\( y = \frac{1}{2} x + 2 \)[/tex]
This equation fits the standard form of a linear function [tex]\( y = mx + b \)[/tex], where [tex]\( m = \frac{1}{2} \)[/tex] and [tex]\( b = 2 \)[/tex]. This indicates a linear relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex], with a slope of [tex]\( \frac{1}{2} \)[/tex] and a y-intercept of 2.
Given this analysis, the equation that represents a linear function is:
[tex]\[ y = \frac{1}{2} x + 2 \][/tex]
So, the answer is:
[tex]\[ \boxed{4} \][/tex]
