The above graph is not a function. In order to be a function, the graph must pass the vertical line test. If you look at x=2, you can see that x=2 is a value on both the green graph and the purple graph. Therefore it fails to pass the vertical line test and is not a function.

This graph is a function. We call this type of function a piecewise (or step) function. This type of function is defined in a certain way for certain x values, and then in a different way for different x values. However, it still passes the vertical line test, so it is a function!

Firstly, this is a function as it passes the vertical line test.

To find the domain, begin at the far left of the graph, starting with the red portion. You can trace along it continuously until you get to the point (-5,-5). At this point, it jumps upward to the green graph. While the y values jump, there is no gap in the x values. You can trace from left to right along the green graph until you get to x=2. Then, the graph jumps up to the purple values. Once again, there is no gap in the x values. The purple graph is continuous and goes onwards to positive infinity. Therefore, the domain of this graph is all real numbers or .

Next, let's start again at the bottom of the graph (in red), this time looking at the y values. We begin at negative infinity and can trace continuously until we get to the point (-5,-5). There is a discontinuity between the red and green graphs' y values. We use the  symbol ("union") to connect the disconnected parts of the graph. The green graph picks up at -1.71 and the y values continue upward until they get to 1.256. Once again, there is a discontinuity in the range, so we again use the union symbol. The y values pick back up at the point (2,4) and continue upwards to infinity. Therefore the range of this function is .

The graph of  is

Keep in mind that you cannot put a negative number under a square root and get a real number answer. If we tried to plug in -2 for x, it would be undefined. That is why the graph only shows positive values of x. 

The other graphs pictured below are , , and .

The graph of  is

.

The other graphs shown are those of , , and .

The graph of  is

The other graphs pictured are , , and .

This graph intersects the x axis only once and the y axis only once, and these happen to coincide at the point (0,0). Therefore the x-intercept is at x=0 and the y-intercept is at y=0. The graph will go on infinitely to the right. However, the leftmost point is (0,0.) Likewise, the graph will go infinitely high, but its lowest point is also (0,0). Therefore, its domain is  and its range is also .

This graph crosses the x axis only once and the y axis only once, and these happen to coincide at the point (0,0). Therefore the x-intercept is at x=0 and the y-intercept is at y=0. The graph will go on infinitely to both the left and the right. Additionally (and despite the fact that it won't happen as quickly), the graph also will go infinitely high in both its positive and negative y values. Therefore, both the domain and range of this graph are .

To simplify this problem we need to find the least common denominator between the two fractions. To do this we look at 5 and at 8. The least common number between these two is 40.

In order to rewrite each fraction in terms of a denominator of 40 we need to muliple as follows:

we are able to mulitply by 8/8 and 5/5 because those fractions are really just 1 written in a different format.

Now using order of opperations we get the following

Now we have a common denominator and can do our addition to get the simplfied number:

In order to be able to find , we must first find the least common denominator. In this case, it is : 

The equation can now be written as:

Solving for , we get: