Functions - SAT Math

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.

On a problem such as this that asks you to match a graph with a function, you need to use process of elimination using the characteristics of the function that you were provided. Here you're told where the zeroes are, meaning that at those points, and , the graph should cross the x-axis. Only one graph meets those criteria, so you have your correct answer.

In this instance, we can use the outcome of the function when our input is 14 to solve for “n” and thus, understand our full function. If

f(14) = -2 and f(x) = nx + 5, then

n(14) + 5 = -2

If we isolate n, we arrive at

n(14) = -7

Be careful… we can’t stop here! The question hasn’t asked us to solve for n, it has asked us for f(2). So, if we know that our function, with n as a known constant, is

, then

, or 4.

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 .

Whenever you're working with a function with an algebraic definition like the one you're given, your job is to plug in the value (in this case ) wherever appears in that definition. This means that your work should look like:

The key here becomes keeping the negative/positive signs in place given all the addition/subtraction and multiplication of negatives in the numerator. Once you've applied the exponents and multiplication in the numerator, you should have:

This then simplifies to , in which the negatives in numerator and denominator cancel, leaving you with just as your answer.

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.

This problem adds a twist to the classic function setup. You should know that whenever you're dealing with a function defined as , your job is to plug in the given input value anywhere there's an term. Often that value is a number, but here the next thing you're told is that , meaning that your input value is .

The steps remain the same, however: just plug in to and you can solve:

This means that , and you know that it should expand to . So you can set up an equation and solve:

Square the parenthetical term to get:

And then distribute the 3:

And you can now subtract and from each side to simplify:

When you divide both sides by 3 the answer should start to become clear:

You cannot get rid of the term so you can see that and . So satisfies both terms, making 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 .

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.

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 .

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 .

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.

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 .