Inverse Parent Functions and Composition of Functions
Description/Explanation/Highlights
Video Description
This video describes parent functions that are inverses of each other and how composition of functions can be used to show that parent functions are inverses of each other.
Steps and Key Points to Remember
To find inverses of parent functions and show that they are inverses, follow these steps:
- In general, inverses of functions are found by switching the x- and y-values of the functions table of values or by reflecting the graph across the y = x line.
- To find an inverse algebraically, the x and y are switched in the original function and it is then solved for y to get the inverse.
- Sometimes restrictions must be added to the function or the inverse to ensure that the inverse of a function is also a function.
- For example, to find the inverse of the quadratic parent function, \(y=x^2\), switch x and y to get \(x=y^2\) and solve for y which requires taking the square root of both sides leaving the inverse as \(y=\sqrt{x}\).
- However, for the inverse to be a function, we must restrict the domain of x to \(x\geq0\) (we can’t take the square root of negative numbers) and the range to \(y\geq0\) to restrict the roots to only the positive roots. Allowing two roots would violate the function rule that allows x to be assigned to one and only one y (vertical line test on the graph).
- Inverses like the inverse of the parent \(y=x^3\) do not require such restrictions so they are simply found by switching x and y to get \(x=y^3\) and taking the cube root of both sides to get the inverse \(y=\sqrt[3]{x}\).
- To graph a function and its inverse, make a table of values and graph the original function. Then, make a second table of values from the first with the x- and y-values switched. Graph this table. This should be a graph of the inverse and should reflect over the y = x line.
- Composition of functions can be used to show that two functions are inverses of each other. To do a composition, simply plug one function into the other and simplify.
- For example, to show that \(y=x^2\) and \(y=\sqrt{x}\) are inverses, let \(f(x)=x^2\) and \(g(x)=\sqrt{x}\). Now lets find \(f\circ g\) or \(f(g(x))\).
- Since \(g(x)=\sqrt{x}\) we put that into the \(f\) function and find \(f(\sqrt{x})=(\sqrt{x})^2=x\). If the result of the composition is x, the new function is an inverse.
- To show that \(f(x)=x^3\) and \(g(x)=\sqrt[3]{x}\) are inverses, we do \(f(g(x))=(\sqrt[3]{x})^3=x\). Since the result of the composition is x they are inverses.
- To show that \(f(x)=2^x\) and \(g(x)=\log_2{x}\) are inverses, we will find \(g(f(x))=\log_2{2^x}\) and using the property of logs that says when the base and argument are the same (in this case 2), then the composition is equal to x and is therefore an inverse.
Here are some key points to keep in mind when identifying inverse of parent functions and using composition of functions to show inverses.
- Inverses of parent functions are found the same way inverses are found for any function.
- To find the inverse from a graph, make a table of values and switch the x- and y-values. Graph the new table to see the graph of the inverse.
- The graph of an inverse reflects over the y = x line.
- To find an inverse algebraically, switch x and y in the original function and solve for y in the new function. The result will be the equation of the inverse.
- When using composition of functions to show that one function is the inverse of another, plug the second function into the x of the first function and simplify. If the result is x, they are inverses.
Video Highlights
- 00:00 Introduction
- 00:18 \(y=x^2\) and \(y=\sqrt{x}\) parent inverses
- 02:11 \(y=x^3\) and \(y=\sqrt[3]{x}\) parent inverses
- 02:48 \(y=2^x\) and \(y=log_2{(x)}\) parent inverses
- 03:33 Using composition to show \(y=x^2\) and \(y=\sqrt{x}\) are inverses
- 05:32 Using composition to show \(y=x^3\) and \(y=\sqrt[3]{x}\) are inverses
- 06:44 Using composition to show \(y=2^x\) and \(y=log_2{(x)}\) are inverses
- 08:38 Conclusion
- To watch this video on YouTube in a new window with clickable highlights, click here