Question

In Exercises \(25-32,\) use parametric graphing to graph \(f, f^{-1},\) and \(y=x\) $$f(x)=\sin ^{-1} x$$

Short answer

Using parametric graphing, the graphs for the function, its inverse, and the line will show the distinct patterns of each respective function: an upward curving shape for the arcsine function, a repeating wave-like pattern for the sine function, and a diagonal straight line through the origin for the linear function.

Step-by-step solution

Graph the original function

First, the function \(f(x) = \sin^{-1} x\) needs to be graphed. A range of values for x is chosen, preferably from -1 to 1, since \(\sin x\) is defined for these values. For each value of x, find the corresponding value of \(f(x)\), then plot these points on the graph. This results in a curve in the first and fourth quadrants, crossing the origin, and with an upwards curvature.

Graph the inverse function

Next, graph the inverse function \(f^{-1}(x) = \sin x\). Choose a range of x values, typically between \(0\) and \(2π\) to capture one full period of the sine function. Calculate the accompanying values of \(f^{-1}(x)\) and plot these on the graph. This will result in a wave-like pattern that oscillates between -1 and 1.

Graph the line

Finally, plot the line \(y = x\). This line forms a diagonal that intersects the origin, and it can be used as a reference to examine the symmetry of the function and its inverse.