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.

When you're given the definition of a function as you are here, , your job to calculate a function is to take the value in parentheses and plug that in for wherever it appears in the definition. Here, qualitatively, you're being told "whatever is, square it and then subtract from that square." That means that:

So:

And for you'd have:

So:

This means that , so the correct answer is .

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 .

If , you can simply set the two items equal: . From there, you can "complete the square" by subtracting the right-hand side and moving it to the left, getting to: . This should look familiar as a common algebraic equation; it factors to . Accordingly, the solution must be .

To solve, you can simply set the output of the function equal to the algebraic expression:

And cross-multiply:

And solve for :

Alternatively, you could have recognized that we have a positive result, so the numerator of our fraction cannot be , and that the values for x greater than 1 would all leave a negative denominator, so that cannot be, either.

This problem tests your familiarity and comfort with function notation. When you're given a definition like , it's important to recognize that is the "input" (whatever they tell you is, you then put that into the equation), and that is the "output" (once you've put your input through the equation, the result is .

Here you're given the function definition and then told that the output, , is equal to . Then you're asked to solve for , which means that you're asked to solve for the input.

So what they're really asking is "for what number, when you take to that power and then subtract the result from , would you end up with ?"

In equation form, that's . Performing the algebra, you can add to each side and subtract from each side.

That gives you: . Knowing your powers of , you should recognize that and , so must be between and .

When you're working with "nested functions" - problems in which you're asked to apply a function to a function, such as here - you should follow classic Order-of-Operations and start with the interior parentheses first. Here that means taking , the inner function, and then using that result as your input for the outer function.

In the definition , the left-hand side of the equation defines your "input" saying "whatever you see in the parentheses where currently is, do to that value what is done to on the right-hand side of the equation." And then the right-hand side of the equation tells you what to do to your input. Here it's saying "take your input and square it, then subtract one."

When you apply that to , you'll do exactly what that says: plug into the spots, meaning you'll take . The result of that is . So .

Now with your input is , so you'll plug in for in . This means that you'll have:

That simplifies to .

When you're given a function definition in the form ... as you are here, your job is then to plug in the value in parentheses anywhere that appears. That means that to solve for you'll just plug in for the in :

And to solve for you would do the same thing, plugging in in place of

To finish the problem, you'll then add 7 + 9 to get the correct answer, 16.

The question gives you that and asks for the value of . The easiest thing to start here is to find the value for where .

You're given that . You can then set this equal to 0 to get . This means that has to equal either or . In order for to equal either of these, must be or . Only is a provided answer choice, so is the correct answer.

Whenever you're working with a function defined as , your job is to take the input value--the value in parentheses--and insert it wherever there's an in the definition. Since here tells you that your input value is , you can plug that into the function:

You're told that so you can set up an equation:

And then you can perform the exponent:

And work to get like terms together:

So , meaning that .