Question

Find the inverse of the function. Then graph the function and its inverse. $$ y=x^3-2 $$

Short answer

The inverse of the function \(y=x^3-2\) is \(y=\sqrt[3]{x+2}\). When these two functions are graphed, they are mirror images of each other reflected over the line \(y=x\).

Step-by-step solution

Find the inverse of the function

First, replace \(y\) with \(x\) and \(x\) with \(y\) in the equation, then solve for \(y\). The original function is \(y=x^3-2\). Therefore, switch \(x\) and \(y\) to get \(x=y^3-2\). Solve for \(y\) to find the inverse, \(y=\sqrt[3]{x+2}\). Discovering the inverse of a function can be performed by just switching \(x\) and \(y\). After that, rearrange the equation to solve for \(y\).

Graph the function and its inverse

Plot the graph of original function \(y=x^3-2\). For this type of cubic function, the graph will increase from left to right, passing through the point (0,-2) which is the intercept. Then plot the graph of its inverse function \(y=\sqrt[3]{x+2}\). For this graph, it is the reflection of the original function graph against the line \(y=x\), passing through (-2,0).

Verify by superimposing the reflection line

Draw the line \(y=x\) on the same system of axes with the original function and its inverse function. If they are mirrored against the \(y=x\) line, it can be concluded that the inverse function was correctly found.

Key concepts

Graphing Functions

When graphing functions, you are essentially visualizing the set of ordered pairs that make up the function. For instance, the function \(y = x^3 - 2\) generates a graph by substituting different x-values into the function, calculating the corresponding y-values, and plotting these points on the plane.
It's important to consider:

• Intercepts: Like the point \((0, -2)\) where the graph intersects the y-axis.
• Increasing or Decreasing Behavior: For \(y = x^3 - 2\), the curve consistently rises from left to right.
• Symmetry and Shape: Cubic functions often exhibit an S-shaped curve.

Graphing aids in understanding the function's general behavior. Especially useful when dealing with more complex equations. You can also check the validity of inverses by using graphs. By drawing the line \(y = x\), you can see if an inverse graph reflects accurately against this line, ensuring the mathematical inversion is correct.

Cubic Functions

Cubic functions are polynomials of degree three, typically represented as \(y = ax^3 + bx^2 + cx + d\). They often produce a distinctive S-shaped curve when graphed. In this exercise, we specifically use the function \(y = x^3 - 2\), which is a simpler form of a cubic function.
Some key points to note about cubic functions include:

• Roots: The solutions of the equation where the function intersects the x-axis.
• Inflection Point: The point where the curvature changes direction, a hallmark of cubic graphs.
• End Behavior: As \(x\) approaches positive or negative infinity, the value of \(y\) does the same, creating the leading S-curve behavior.

Due to these properties, cubic functions can model complex scenarios in physics, economics, and natural phenomena. Understanding their unique characteristics can help you simplify complex problems by identifying key behaviors of these intriguing polynomials.

Function Transformations

Function transformations allow you to alter the graph of a function in predictable ways. This concept is critical for graphing functions like \(y = x^3 - 2\) and its inverse without fully recalculating everything.
Here's a simple breakdown of various transformations:

• Vertical Shifts: Adding or subtracting a constant moves the graph up or down. In \(y = x^3 - 2\), the graph is shifted downward by 2 units.
• Reflections: Graphs flip over a designated axis. The inverse function \(y = \sqrt[3]{x+2}\) is a reflection across the line \(y = x\).
• Horizontal Shifts: Changes in the y-value based on adjustments inside the function's argument, such as \(x + 2\), move the graph left or right.

Mastering these can speed up understanding and plotting, letting you easily shift, reflect, and scale graphs as required. Visualizing these shifts can help confirm if your inverse function is plotted correctly by seeing if it aligns as expected with transformations.