Reflect on the concept of composite and inverse functions. What concepts (only the names) did you need to accommodate these concepts in your mind? What are the simplest composite and inverse functions you can imagine? In your day to day, is there any occurring fact that can be interpreted as composite and inverse functions? What strategy are you using to get the graph of composite and inverse functions?
To understand the concepts of composite and universe functions, I had to familiarize myself with a few related ideas. First, I needed to understand the concept of a function itself. A function is a mathematical relationship between two sets of values, where each input value (the domain) corresponds to exactly one output value (the range) (Abramson, 2021). This understanding was crucial in grasping the concept of composite and inverse functions.
Next, I had to understand what it means for functions to be combined or composed. In the case of composite functions, it involves taking the output of one function and using it as the input for another function. The result is a new function that represents the composition of the two original functions. This idea of combining functions is similar to how we can combine different operations in mathematics to create more complex expressions.
Regarding inverse functions, I needed to comprehend the concept of an inverse relationship. An inverse function is essentially the “reverse” of a given function. It undoes the effect of the original function, taking the output values and mapping them back to their corresponding input values (Abramson, 2021). In other words, if we apply a function to a value and then apply its inverse function to the result, we should end up with the original value.
Now, let’s consider some simple examples of composite and inverse functions. A straightforward composite function could be the combination of two linear functions, such as f(x) = 2x + 3 and g(x) = 4x – 1. The composite function (g ∘ f)(x) would involve plugging the expression for f(x) into g(x), resulting in (g ∘ f)(x) = g(f(x)) = 4(2x + 3) – 1 = 8x + 11.
As for inverse functions, a simple example could be the function f(x) = 2x. Its inverse function, denoted as f^(-1)(x), would be f^(-1)(x) = x/2. Applying the function f(x) to a value and then applying its inverse function should yield the original value. For instance, if we take f(4) = 2(4) = 8 and then apply f^(-1)(x) to 8, we get f^(-1)(8) = 8/2 = 4, which is the original input.
In our day-to-day lives, there are occurrences that can be interpreted as composite and inverse functions. For example, let’s consider the process of converting temperatures from Celsius to Fahrenheit and vice versa. We can view this as a composite function, where the Celsius-to-Fahrenheit conversion is one function and the Fahrenheit-to-Celsius conversion is another function. By applying these functions in sequence, we can convert temperatures back and forth between the two scales. Also, if you drive a car at a certain speed for a certain amount of time, the distance you travel can be seen as a composite function. The speed function and time function combine to give you the distance function. Similarly, if you want to convert Celsius to Fahrenheit and then back to Celsius, you would be using inverse functions.
When it comes to graphing composite and inverse functions, I rely on a few strategies. For composite functions, I start by graphing the individual component functions and then combine their graphs according to the composition rules. This helps me visualize how the output of one function becomes the input of another. As for inverse functions, I use the reflection method. I plot points on the original function and then reflect them across the line y = x to obtain the corresponding points on the inverse function. Connecting these points helps me sketch the graph of the inverse function.
To sum up, the concepts of composite and inverse functions require an understanding of functions, the combination of functions, and the notion of inverse relationships. Simple examples of composite and inverse functions involve linear functions, and in our daily lives, we can interpret temperature conversions as composite and inverse functions. Graphing composite functions involves combining the graphs of individual functions, while graphing inverse functions can be done using the reflection method.
References
Abramson, J. (2021). Algebra and trigonometry (2nd ed.). OpenStax, TX: Rice University. Retrieved from https://openstax.org/details/books/algebra-and-trigonometry-2e