To determine if the relation [tex]\( y = 4x - 1 \)[/tex] is a function, follow these steps:
1. Graph the Relation:
Let's plot the linear equation [tex]\( y = 4x - 1 \)[/tex]. This equation represents a straight line where the slope is 4 and the y-intercept is -1.
- The y-intercept means that when [tex]\( x = 0 \)[/tex], [tex]\( y = -1 \)[/tex]. So, the point [tex]\( (0, -1) \)[/tex] is on the graph.
- The slope of 4 indicates that for each unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 4 units.
To graph this, you could also find another point by choosing a value for [tex]\( x \)[/tex]. For example, let [tex]\( x = 1 \)[/tex]:
[tex]\[
y = 4(1) - 1 = 4 - 1 = 3
\][/tex]
So, the point [tex]\( (1, 3) \)[/tex] is also on the graph.
Draw a straight line passing through these points [tex]\( (0, -1) \)[/tex] and [tex]\( (1, 3) \)[/tex].
2. Apply the Vertical Line Test:
The vertical line test is a method to determine if a relation is a function. According to the test:
- Draw vertical lines (lines parallel to the y-axis) across the graph.
- If any vertical line intersects the graph in more than one point, the relation is not a function.
- If every vertical line intersects the graph at most once, the relation is a function.
For the straight line represented by [tex]\( y = 4x - 1 \)[/tex]:
- No matter where you draw a vertical line, it will only intersect the straight line at exactly one point.
Therefore, every vertical line drawn on this graph will meet the graph at exactly one point.
3. Conclusion:
Since the vertical line test is satisfied, the relation [tex]\( y = 4x - 1 \)[/tex] is indeed a function.
So, we confirm that the correct answer is:
[tex]\[
\boxed{1}
\][/tex]
