### All SAT Mathematics Resources

## Example Questions

### Example Question #11 : Working With Function Notation

If and , what is the value of ?

**Possible Answers:**

**Correct answer:**

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 .

### Example Question #11 : Working With Function Notation

If , what is ?

**Possible Answers:**

-6

9

5

8

**Correct answer:**

8

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.