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Inverse functions are used to reverse the effect of a function. If you start with a value, apply a function to it, and then apply the inverse function, you should end up with your starting value. To find the inverse of a function like \( f(x) = x^3 - 1 \), you swap the x and y values and solve for the new y. This means if you have \( y = x^3 - 1 \), you solve it as \( x = \sqrt[3]{y + 1} \). So, the inverse function is \( f^{-1}(x) = \sqrt[3]{x + 1} \).
In practice, this helps us understand how values map between different spaces. Plot points for the original function and then their corresponding inverses to see this relationship come to life.

Graphing functions requires plotting points from a table of x and y values. For the function \( f(x) = x^3 - 1 \), you choose some x-values, compute the corresponding y-values (e.g., if \( x = -2 \), then \( f(-2) = -8 - 1 = -9 \)), and plot these points on a graph.
Once you have enough points, connect them smoothly to reveal the curve of the function. Each x-value corresponds to exactly one y-value. This visual representation helps us understand how the function behaves, such as where it increases, decreases, or stays constant.

The coordinate plane is a two-dimensional space with a horizontal axis (x-axis) and a vertical axis (y-axis). Each point on the plane is represented by a pair of coordinates (x, y). The coordinate plane helps us visualize functions and their inverses. For example, to plot \( f(x) = x^3 - 1 \), we place points corresponding to certain x-values and their resultant y-values.
When graphing an inverse function like \( f^{-1}(x) = \sqrt[3]{x + 1} \), choose x-values again and compute their y-values. Plot these on the same plane. To verify the inverse relationship, you’ll see points switch places (a, b) -> (b, a) and the graphs will be symmetric around the line \( y = x \). This shows that one function undoes the effect of the other.