To determine which equation represents a linear function, we need to recall the general form of a linear equation. A linear function is one that can be written in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] and [tex]\( b \)[/tex] are constants, and [tex]\( x \)[/tex] is the variable. This indicates a straight-line graph.
Let's examine each given equation one by one:
1. [tex]\( y = x^2 + 3 \)[/tex]
- This is not a linear equation because it contains [tex]\( x^2 \)[/tex], which makes it a quadratic equation. The graph of [tex]\( y = x^2 + 3 \)[/tex] is a parabola, not a straight line.
2. [tex]\( x = y^2 - 2 \)[/tex]
- This equation expresses [tex]\( x \)[/tex] as a function of [tex]\( y \)[/tex] and includes [tex]\( y^2 \)[/tex]. Therefore, it is also not linear since it represents a quadratic relationship. The graph of [tex]\( x = y^2 - 2 \)[/tex] is a parabolic shape when viewed in terms of [tex]\( y \)[/tex].
3. [tex]\( x = 1 \)[/tex]
- This is an equation representing a vertical line where the value of [tex]\( x \)[/tex] is always 1, regardless of [tex]\( y \)[/tex]. While it is a straight line, we generally express linear functions as [tex]\( y \)[/tex] being dependent on [tex]\( x \)[/tex], not the other way around. Therefore, we won't consider it under the typical form [tex]\( y = mx + b \)[/tex].
4. [tex]\( y = \frac{1}{2} x + 2 \)[/tex]
- This equation is in the form [tex]\( y = mx + b \)[/tex] with [tex]\( m = \frac{1}{2} \)[/tex] and [tex]\( b = 2 \)[/tex]. It perfectly fits the definition of a linear equation. The graph of this equation is a straight line with a slope of [tex]\( \frac{1}{2} \)[/tex] and a y-intercept of 2.
Given our analysis, the equation that represents a linear function is:
[tex]\[ y = \frac{1}{2} x + 2 \][/tex]
So, the correct choice is:
[tex]\[ 4 \][/tex]
