# SAT Mathematics : Working with Function Notation

## Example Questions

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

If  and , what is the value of ?

Explanation:

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 ?

-6

9

5

8

8

Explanation:

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: