There is a graph of f(x) and g(x). They intersect at two points. Michael says that the values of (x)=g(x) are the ordered pairs where the functions intersect. Find Michael's mistake and explain what he misunderstood.
Eric isn't sure why a solution to a system of inequalities cannot be on a dotted line of a graph. Explain why points on the dotted line of an inequality can't be in the solution set.
Huh? y = f(x) is one function
y = g(x) is another function
They , f and g, are the same for two values of x
Those points are where y = f(of that x) and same y = g(of that same x)
so at those points
y = f(x) = g(x)
and the points,( x,y) are (x ,f(x)) and (x,g(x)) and f(x)=g(x)
*sorry, Michaels problem was supposed to say f(x)=g(x)
To find Michael's mistake in his statement about the intersections of the functions f(x) and g(x), let's take a closer look at his misunderstanding. According to Michael, the values of (x)=g(x) are the ordered pairs where the functions intersect. However, this is not entirely accurate.
When two functions intersect, it means that there are points on their respective graphs that have the same y-coordinate (or f(x)-value in this case). Therefore, to find the points of intersection, we need to compare the y-values of the two functions at each corresponding x-value.
In other words, to find the ordered pairs where the functions f(x) and g(x) intersect, we need to solve the equation f(x) = g(x) for x. This will give us the x-values of the intersections, which we can then plug into either f(x) or g(x) to find the corresponding y-values.
So Michael's mistake lies in assuming that the ordered pairs where the functions intersect are solely determined by the condition (x) = g(x). But in reality, the points of intersection are determined by (x, y) values where f(x) = g(x).
Shifting our attention to Eric's question about why a solution to a system of inequalities cannot be on a dotted line of a graph, let's dive into the reason behind this limitation.
In the context of inequalities represented graphically, a solid line is used to indicate that the points on the line itself are included in the solution set. On the other hand, a dotted line signifies that the points on the line itself are not included in the solution set.
The distinction between a solid line and a dotted line serves as a boundary in the graphical representation. For example, consider a linear inequality such as y ≥ 2x. If we plot this inequality on a graph, the points on the corresponding line y = 2x would be included if it were a solid line, but not if it were a dotted line.
The reason behind this is that the solution to an inequality involves determining the set of points that satisfy the given condition. In cases where a dotted line is used, it implies that the equation is not satisfied at that specific point. Therefore, points on the dotted line are not viable solutions that meet the specified inequality criteria.
To summarize, points on a dotted line of an inequality cannot be in the solution set because they do not satisfy the inequality condition. Only points either above or below the dotted line, as determined by the direction of the inequality symbol, are valid solutions.