As with many functions problems, this problem is testing whether you can follow directions. You are given two functions and then asked for the value of a nested function, . Remember when you are evaluating functions, that you must always start from the innermost function and work outward since you must always follow the order of operations. So, in this case, the problem is asking you to find the value of  and then to plug the resulting value into .

Plugging in for  is simple: you simply take whatever is the parentheses (in this case, ) and plug it in for  in the function. Your function then transforms from  to .

Your next step will be similar: you just need to take what is in parentheses  and plug that value into the function  every place you see an . This means that  becomes . notice that the entire value  is plugged into the function, not just . If you expand this function using your knowledge of perfect squares, it becomes: , which in turn becomes .

Notice that you could also plug in a number for . If you plug in , the  becomes . As above, remember to start from the inside and work out.  becomes . 

The function  can then be rewritten as . If you then plug in  to the function , you get: .

In order to finish this problem, you simply need to plug your original value for ,  into each answer choice, and look for .

Function questions tend to derive most of their difficulty from the abstract function notation itself. So being comfortable with approaching function notation is most of the battle. When you see function notation such as , keep in mind that  is the "input" (whatever they tell you  is, put that into the equation), and that  is the "output" (once you've put your input through the equation, the result is the value of ).

So when you're given , what the problem is really saying is that "whatever we put in the parentheses of , plug that value in wherever you see an  in . Which means you'll take the input value, square it, subtract the product of the input value and two, add one to that, and then take the square root of the whole thing. With 9, that looks like:

You can then simplify the math underneath the radical to get:

And since you know that  you have your answer.

Some SAT questions ask you to interpret the graph of a function, even if you are not given the algebraic notation for that function. For these, recognize that when a function is graphed, the value of the function is shown on the y-axis (the vertical height) and the input values are displayed on the x-axis (the horizontal). So when this question asks you for the maximum of the function, look for the highest spot on the graph. That takes place at the point (0, 6). Since the question asks for the x-value at which the function is maximized, you'll then use x = 0 as your answer.

This problem asks you to interpret the graph of a function. As you can see, revolutions per minute are graphed on the y-axis, so to find her highest RPM you're looking for the highest point on the graph. But the question asks for how many minutes she was into the workout at that point, and minutes are on the x-axis. So when you find her highest point, then scan down to the x-axis to see the time, which is at approximately 21 minutes.

When approaching this problem, it is important to know that in functions, , graphed on the y-axis, is the value of the function, the output value. And  is the input value. So when you're asked to find the minimum and maximum of a function, you're looking on the y-axis for the lowest (minimum) and highest (maximum) points. Those points here occur at (-1, 7) for the maximum (that's where the function has a value of 7) and at (3, 1) for the minimum (that's where the function has a value of 1). Since the question asks for the difference in the coordinates at those points, you'll take those coordinates of -1 and 3 and. The difference is then 3 - (-1) = 4.

An important thing to know about functions is that f(x) is defined as the "function" itself, where x is an input value that goes into the function. So when you're asked for where a function is at its minimum, you're looking on the y-axis for the lowest point.  Here that point is at (10, 1). The question asks for x value, so the correct answer is 10.

An important consideration when you're interpreting graphs of functions is that the value of the function itself - in this case the function g(x) - is marked on the y-axis (the vertical axis). The x-value is an input value to the function, and the y-value is the value of the function itself. So here you're looking for where the graph crosses the x-axis, at which point y=0.  This happens at the point (1, 0). And since the question asks you for the value of x at this point, you'll then provide the correct answer, x = 1.

As you can see from the graph, the team gradually increased its probability of winning during the first half of the game, then that probability began to drop gradually in the second half, with two sharp drops until the probability reached 0.  This demonstrates that the team lost the game - at the end, its probability of winning was zero. While the team did increase its probability of winning throughout the first half, it did so gradually, so the idea that it was "significantly better than its opponent" is not necessarily true.

Importantly when you are dealing with functions, you should know that the value of the function itself is shown on the y-axis (the vertical axis) and the x-axis shows you the input value of x that leads to the value of the function.  So here when you're asked for the minimum value of the function, you should look for the lowest point on the graph.  That appears at approximately (3, -1).  The question asks for the x-value at that point, so the correct answer is 3.

When you're interpreting graphed functions, it is important to recognize that the value of the function itself is shown on the y-axis (the vertical axis) while the horizontal axis displays the x-values that are inputs to the function. So here when you're asked for the minimum value of the function, you're looking for the lowest point, which happens at the point (-5, 2). And since the value of the function is the y-coordinate at that point, you can answer 2 as the correct answer.