Domain and range

Question

Answer 1

The concept of inverse functions (f-1) is a lot simpler than it may seem: we want to take our current expression and rearrange it so that we can plug in outputs (y) and receive pure inputs (x), rather than plugging in inputs and receiving pure outputs like we are with the current form of our expression.

For example, let's say I have a function y = 3x. I can plug in input to x and get a value right away for y; when I put in x=1, we receive y=3; for x=2, y=6, and so on. But what if I wanted to plug in a value for y, say, y=3, and know what x comes out? Well, then we get 3 = 3x...and that's not so straightforward. We still don't know what x is.

So, how do we go about this? Let's look at an example of y = 2x+4.

If we're looking at a graph, we can simply transform it so that x points up and y points right:

Notice this is the same function: we cross at x = -2 and y = 4. We are just looking at it differently, so that x is now vertical (which is how we usually put our output, usually y) and y is now horizontal (which is how we usually put our input, usually x).

All we want to do know is say, "Ok, our inputs from before, what we called x, are now our outputs, and our outputs, what we called y, are now our inputs. So, all we need to do is rewrite x as y and y as x to represent this.

This is our inverse function, f-1! Let's write it over our first graph to compare the two.

They look similar, don't they? Let's rotate to see just how similar they are:

Pleasing to the eye, right? They're symmetrical around y=x: if you look to the right of y=x at a certain distance to the right, the other line is that same distance from y=x going to the left. Because we wanted to invert our x and y, so that our inputs are outputs and vice versa, they're both symmetrical about y=x, because for this unique function, the input x is always exactly equal to the output y.

So, we can do this for any inverse function with an input x and an output y. If we draw our y=x line and our function, we can always find the inverse function f-1 by drawing the same line, rotating our paper a bit so y=x is vertical, and redrawing the line, so that it's flipped and symmetrical to y=x.

Answer 2

The graph of function, f(x), and the graph of its inverse function, f^–1(x), are related by reflection across the line y = x (the dotted line in the graphs of your answer choices). For which of the four graph choices( A, B, C, or D) is this true?

Hope this helps. Add a comment if you have any questions.