Composition and Transformation of Functions

Now, let\’s begin our discussion of composition of functions and finding the inverse of a function:

Computing the composition of two functions (Section 1.6) and computing the inverse of a function (Section 1.8), are two of the most important topics we are covering in this module. These algebraic procedures are important in theory and in practice and are relevant in the study of calculus. This discussion activity will help you to become proficient in these skills.

Choose one of the two options:

Create and post two functions, f(x) and g(x), of the types we have considered so far in this course. Make the functions challenging, but not beyond what is covered in the textbook; avoid very simple examples.
Create and post a function, f(x), again, of a type we have already considered in this course.

In this module, we will investigate how the graph/behavior of a given function changes when a constant is added to it, or multiplied by it, or if we replace the independent variable by the variable plus a constant. These are called transformations of a function and are central to really understanding how functions behave. We will then look at how we can add, subtract, multiply, and divide functions, analogous to how we perform such operations with numbers in arithmetic. This is called combining functions. Next, and somewhat related, is evaluating a function at another function (instead of a specific value of the independent variable). This interesting and important topic is called the composition of functions. As we will see, this naturally leads to the idea of an inverse of a given function, which has direct applications in calculus, and in useful, algebraic real-world applications, too.

Finally, we will consider some various, related topics, such as how to determine certain properties of circles based on the standard form of the circle, and how to calculate the distance and midpoint values between any two pairs of points in the Cartesian coordinate system.

All of these topics help us to understand the nature of functions, which is important in properly modeling real-world phenomena.

Operant conditioning has many uses in our everyday lives. Behavior modification uses mostly operant methods in an attempt to change undesirable behaviors such as bad habits. Similar principles are also used in behavioral therapy for a wide range of psychological disorders. Closer to home, operant conditioning can be used to modify the behavior of children and to train pets. In this exercise, we will look at how an animal can be trained by using reinforcement of desired behavior.

As you complete this exercise, consider the following:

How can the principles of operant conditioning be applied to changing behavior?How does reinforcement contribute to behavior change?
By the end of this activity, you will be able to:

Explain the principles and processes involved in operant conditioning

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