This applet shows the inverse relation of a function.[br][br]The inverse of [math]y = f(x)[/math] is a relation [math]x = f^{-1}(y)[/math]. Graphically, the inverse relation is obtained by reflecting the graph of [math]y = f(x)[/math] about the line [math]y = x[/math].
Enter the rule for a function f(x) in the textbox at bottom-left. Click 'Show points' to display a point on the x-axis, and the point(s) corresponding to [math]f^{-1}(x)[/math]. Drag the blue point to change x.[br][br]What do you get as you drag x along the axis?[br][br]Click 'Show inverse' to display the entire inverse relation. Is it a function? You can click 'Vertical line test' to provide a vertical line which might help you decide.
Is the inverse of [math]y=x^2[/math] a function? Does it pass the Vertical Line Test?
Can you change the domain of [math]y=x^2[/math] to make its inverse a function? Explain
Change [math]f\left(x\right)[/math] to [math]x^3[/math] in the applet. You can either use the carrot key (shift + 6) or tap the keyboard icon in the input field to type in the function.
Is the inverse of [math]f\left(x\right)=x^3[/math] a function? Does it pass the Vertical Line Test?
Change [math]f\left(x\right)[/math] to [math]x^4[/math] in the applet. You can either use the carrot key (shift + 6) or tap the keyboard icon in the input field to type in the function
Is the inverse of [math]y=x^4[/math] a function? Does it pass the Vertical Line Test?
Can you change the domain of [math]y=x^2[/math] to make its inverse a function? Explain
Try other power functions. Is there a way to tell whether a power function's inverse is itself a function? Either by looking at the equation or devising a fool-proof test? Exaplain.