20.8 Inverse functions
Functions are relations where the x values will only have one y element associated with it. So, if we’re given a set of ordered pairs (1,0) (2,5), (3,15), and (4,20) then we know that this illustrates a function. But if we’re given (1,0), (1,5) (2,15) (3,20) then we would know that this isn’t a function. The x values are referred to as the domain of the function and the y values is the range.
This chapter will have eight parts that would discuss the basics things that we need to know about functions. The first part will be all about the function notation. A function is written through the function notation f(x), say for example f(x) = 2x +5. F(x) simple means that for a particular value of x, the equation will be equal to a certain value. So if x is 1, then the function above will be f(1) = 2(1) + 3 which would give you the y value 3, thus your ordered pair would be (1,1).
The proceeding parts of this chapter from section 2 to 6 will be all about operation with function. We will learn how to add functions like in the case of (f +g) (x), subtract functions like in (fg)(x), multiply function like in (f x g) (x) and divide function like (f/g)(x).
Apart from learning about the operation with functions, we will also look at the composition of functions and the inverse functions. For the composition of function like (f ? g)(x), where f(x) =x + 1 and $g(x) = x^2 + 1$, then we will need to substitute the value of x for g(x), so this will become $f(x^2 + 1) = x + 1$, which is equal to $(x^2 + 1) + 1$. Simplifying that, we get $x^2 + 2$.
For the inverse function like $f^1(x)$, we will see that this is just the reverse of f(x). In this chapter we will learn to find the domain and range of the inverse functions and also learn how to graph them. Graphing inverse function is better understood once we have mastered how the graph the reverse of the inverse functions. We can use an online functions graphing calculator to check if we are able to graph them correctly.
Understanding the basic concepts of the relations and functions is very important for the next chapters. There are several exercises that would illustrate what we have discussed earlier and would help us understand each concept covered by the chapter.
Inverse functions
Lessons

a)
Sketch the graph of the inverse $y = {f^{  1}}\left( x \right)$ in 2 ways:
i) by reflecting $f\left( x \right)$ in the line $y = x$
ii) by switching the x and y coordinates for each point on $f\left( x \right)$ 
b)
Is $f\left( x \right)$ a function?
Is ${f^{  1}}\left( x \right)$ a function?


2.
Consider the quadratic function: $f(x) = (x+4)^2 + 2$

3.
Determine the equation of the inverse.
Algebraically determine the equation of the inverse ${f^{  1}}\left( x \right)$, given: