Introduction

This week we are studying lessons 1.6 and 4.1. We won’t have a big test on chapter 1 – instead we’ll do a couple quizzes over the next two weeks. There will be a test at the end of chapter 4 that will cover topics from both chapters.

Lesson 1.6

Lesson 1.6 is on the algebra of functions. You will learn how to add, subtract, multiply and divide functions, and find their resulting domains. This is pretty straightforward – you just add, subtract, multiply, or divide the two equations and simplify or add like terms as needed.  You will also learn how to find the composition of two functions. This means that you put one function inside the other; instead of evaluating a function at an x-value, you evaluate it at another function. For example, if f(x) = x+1 and g(x) = 2x/5, then f composed with g is written f ο g (the little circle means compose the two functions). f ο g = f [g(x)] = f [2x/5] = 2x/5 + 1. So basically, you put in the inside function wherever you see an x.

Lesson 4.1

Lesson 4.1 is on inverse functions. The inverse of a function is the function that will “undo” the function.

So if f(x) = 2x +5, f(x) is a function that multiplies the input by 2 and then adds 5. The “undoer” function would subtract 5 and then divide the result by 2. So the inverse function is f ¯¹(x) = (x-5)/2. f  with a little -1 superscript means f inverse.

Now try this:

1. Find f(1). (the output of f when you input 1)
2. Plug that value into f ¯¹(x).
3. Notice that the output is your original input.

Some things to note about inverse functions:

• If you put an output (y-value) of the original function into the inverse function, you will get out your original input (x-value).
• The x-values in f(x) are the y-values in f ¯¹(x), and the y-values in f(x) are the x-values in f ¯¹(x).
• The graph of a function and its inverse are reflections of eachother across the line y = x.
• The composition of a function and it’s inverse equals x. So f [f ¯¹(x) ] = x. This is because you are doing something to x, then undoing that process, so you just get out x.

Assignment  

1. Watch the video on section 1.6.

2. Watch the video on section 4.1.

3. Complete the homework assignment.