A curve in the XY plane is the group of some function f if and only if no vertical line intersects the curve more than once.
For example, the parabola x=y^2-2 shown in Figure (a) is not the graph of a function x of because, as you can see, there are vertical lines that intersect the parabola twice. The parabola, however, does contain the graphs of two functions of x . Notice that the equation x=y^2-2 implies y^2=x+2 , so
y=Sqrt(x+2) and -[Sqrt(x+2)] .
Thus the upper and lower halves of the parabola are the graphs of the functions f(x)=Sqrt(x+2) and g(x)= -[Sqrt(x+2)] [See Figures (b) and (c).] We observe that if we reverse the roles of x and y , then the equation
x = h(y) =y^2-2 does define x as a function of y (with y as the independent variable and x as the dependent variable) and the parabola now appears as the graph of the function h .
For example, the parabola x=y^2-2 shown in Figure (a) is not the graph of a function x of because, as you can see, there are vertical lines that intersect the parabola twice. The parabola, however, does contain the graphs of two functions of x . Notice that the equation x=y^2-2 implies y^2=x+2 , so
y=Sqrt(x+2) and -[Sqrt(x+2)] .
Thus the upper and lower halves of the parabola are the graphs of the functions f(x)=Sqrt(x+2) and g(x)= -[Sqrt(x+2)] [See Figures (b) and (c).] We observe that if we reverse the roles of x and y , then the equation
x = h(y) =y^2-2 does define x as a function of y (with y as the independent variable and x as the dependent variable) and the parabola now appears as the graph of the function h .
No comments:
Post a Comment