Function questions are some of the trickiest question types you'll see on SAT Math. They can look intimidating, require higher-level math knowledge, and often take numerous steps to solve. Not to mention, a lot of students haven't had a lot of experience solving functions.
Fortunately, there's a way to make SAT function questions much less intimidating. Learning their basic properties and the types of function questions you'll see on the SAT will go a long way to helping you tackle these questions with confidence. And that's important because you'll likely see ten or more function questions when you take the SAT.
Read on to learn what functions are, the four types of functions SAT questions you'll see, step-by-step strategies for solving them, and tips for making any function question easier to solve.
What Are Functions?
Before we dive into actual functions SAT questions, let's go over some basic information about functions. Having this solid foundation of knowledge will make answering SAT function questions easier.
In the most basic sense, a function is a rule that takes an input, does something to it, and gives you a specific output. Think of it like a vending machine: you press a button (the input), and you get a snack (the output). Every time you hit the same button, you'll get the same snack, but if you change the input by pressing a different button, you'll get a different output/snack.
Each input (x value) can produce only one output (y value), but one output can have multiple inputs. In other words, multiple inputs may produce the same output. Going back to the vending machine, if you hit button B12, you'll always get, say, Cool Ranch Doritos. You won't suddenly get Peanut M&Ms if you keep hitting B12. That one input will always lead to the same output. However, because Cool Ranch Doritos are so delicious, many vending machines have multiple rows of them to make sure supply is high enough. So you may also be able to hit, say C4 and also get Cool Ranch Doritos. The output (the Doritos) can be gotten through multiple inputs (buttons B12 and C4). This means that a function graph can have potentially many x-intercepts, but only one y-intercept. This is because when the input is x=0, there can only be one output, or y value.
Functions are commonly written in function notation: f(x) = _______. Here, f is the name of the function, $x$ is the input, and whatever comes after the equals sign tells you what happens to that input. For example, if you see $f(x) = x + 3$, that means whatever number you plug in for $x$, the function will add 3 to it. So:
$f(2) = 2 + 3 = 5$
$f(10) = 10 + 3 = 13$
Note: we can call our function other names than f. The above function is called f, but you may see functions written as h(x), g(x), r(x), or anything else.
Take this example: $g(x) = 12(x-7) + 4$
- "g" is the name of the function
- "x" is the input
- "12(x-7) + 4" gives us the output once we plug in a value for x
An ordered pair is the coupling of a particular input with its output. For the above function, with an input of 10, the equation would look like:
$g(10)=12(10-7) + 4$
$=12(3) + 4$
$= 36 + 4$
$= 40$
So our ordered pair is (10, 40). Ordered pairs also act as coordinates, and we can use them to graph a function which is what we're discussing next.
Graphing Functions
Functions can always be graphed, and different kinds of functions will produce different looking graphs. On a standard coordinate graph with axes of $x$ and $y$, the input of the graph will be the $x$ value and the output will be the $y$ value.
As a reminder, a function graph can have potentially many x-intercepts, but it will only ever have one y-intercept.
Linear Functions
A linear function makes a graph of a straight line. The graph just above is an example of a linear function. In linear functions, x is never raised to a higher power than 1 (such as being squared, cubed, etc.). Equations of linear functions look like the following examples:
$f(x) = x +14$
$g(x) = 3$
$r(x) = 5x -11$
Quadratic Functions
A quadratic function makes a graph of a parabola, and the output variable will always be squared. Remember that $(2)^2$ and $(−2)^2$ both equal 4. That means quadratic function SAT questions have two input values—a positive and a negative—that give us the same output value. That's why they have the parabolic curve when graphed.
A quadratic function is often written as:
$f(x)=ax^2+bx+c$
The $a$ value tells us how the parabola is shaped and the direction in which it opens.
A positive $a$ gives us a parabola that opens upwards, like the above graph.
A negative $a$ gives us a parabola that opens downwards, like this graph:
The larger the $a$ value is, the "skinnier" the parabola will be.
The $b$ value tells us where the vertex of the parabola is, to the left or right of the origin.
A positive $b$ puts the vertex of the parabola left of the origin:
A negative b puts the vertex of the parabola right of the origin:
The $c$ value gives us the y-intercept of the parabola. This is wherever the graph hits the y-axis. In the graph below, the red dot marks the y-intercept:
Note that when $b=0$, the y-intercept will also be the location of the vertex of the parabola, like this:
Function Questions on the SAT: 4 Examples
Now let's move on to what function questions will look like on the SAT. In general, there are four types of SAT function questions. Below we explain each of them and work through an example. The examples all come from official practice SAT exams.
#1: Function Equations
These are the most common function problems you'll see on the SAT; expect at least six of them on your exam. A function equation problem will give you a function in equation form and then ask you to use one or more inputs to find the output (or elements of the output).
Example:
Explanation:
This is a grid-in problem, one where you aren't given answer choices but need to enter your own answer.
In order to find a particular output, we must plug in the given input for $x$ into the equation. Since the graph of $y = g(x)$ passes through (24,0), that means that $g(24)=0$. We are told that the point (24,0) is on the graph of $y=g(x)$. This means that when $x=24$, $g(24)=0$. That means we want to substitute $x=24$ into the function:
$g(24)=(24+14)(t−24)$
Because $g(24)=0$, we set the equation equal to zero:
$(24+14)(t−24)=0$
$38(t-24)=0$
For this equation to be true, either $38=0$ (which isn't possible) or $t−24 = 0$. So now we know $t =24$.
Substituting 24 for $t$ into the equation gives us $g(x)=(x + 14)(24-x)$. The value of $g(0)$ can be found by substituting 0 for $x$ in the equation:
$g(0) =(0 + 14)(24-0)$
$g(0) = (14)(24)$
$g(0)= 336$.
336 is the correct answer.
#2: Function Tables
The second way you may see a function is in a table. For these questions, you'll either be given a table of values for inputs and outputs and asked to find the equation of the function, or (like the example below), you'll be given a function equation and be asked which table has the correct corresponding values for the function.
Example:
Explanation:
Let's start with what we know. The function is $h(x) = x^2 - 3$. There are four tables, and each has $x$ values of 1, 2, and 3.
Oftentimes the best strategy for these types of questions is to plug in answers. This way, we don't have to actually find the equation on our own—we can just test which answer choices match the inputs and outputs we are given in our table.
Substituting 1 for $x$ in the equation $h(x) = x^2 - 3$ gives us:
$h(1) = 1^2 - 3$
$= 1-3$
$= -2$.
So when $x$ is 1, $h(x)= -2$. Based on that, we already know Choices A and C are incorrect.
We're down to Choices B and D. They both have the same $h(x)$ value when $x=2$, so let's substitute 3 for $x$:
$h(3) = 3^2 -3$
$= 9-3$
$=6$
Choice B is the only one with the correct answers for both $x=1$ and $x=3$, so it's the correct answer!
#3: Function Graphs
These questions will generally ask you to identify specific elements of an already-graphed function or have you find the equation of the function from a graph. Function graph questions may look intimidating, but they're often the easiest of the SAT function questions you'll see. Just remember that $x$ is your input and that your equation is your output, $y$. Function graph questions are also the least common type of function problem on the SAT. At most, you'll see two of them, and it's more likely to only have one or even zero on your test.
Example:
Explanation:
By definition, y-intercept of a graph is the point where the graph intersects the y-axis. The graph of function shown intersects the y-axis at (0, -4). Therefore, Choice B is the correct answer. Easy!
#4: Function Word Problems
This is one of the newer SAT function question types, and you can expect to see about 2 to 3 of these on the SAT. For these problems, you won't be working out equations or making calculations. A function will be given, and you'll need to choose the "best interpretation" of one part of it. Instead of testing your hard math skills, it tests your understanding of functions and what different parts of them represent.
Example:
Explanation:
We know that the function $f(t)= 14t + 9$ gives the estimated length of a plant and $t$ represents the number of months ago the plant was purchased.
Going back to what we know about functions, when a function has the form $f(t)=mt + b$, $b$ represents the value of $f(0)$, which is the same as the value of $f(t)$ when $t=0$. The $b$ in this example is 9. Therefore, the 9 represents the value of $f(t)$ when $t$ (the number of months ago the plant was purchased) is 0. $F(t)$ is the length of a plant, so when the plant was purchased 0 months ago (i.e. just purchased) the plant was nine inches long. It grows longer each month.
Therefore, the best interpretation of 9 is that the vine was 9 inches when Tavon purchased it, or Choice D.
Choice A is incorrect because there's nothing in the function that represents how long Tavon will keep the vine. Choice B is incorrect because, based on the function, the vine is expected to grow 14 inches each month, not 9 inches. And Choice C is incorrect because there's nothing in the function that indicates a maximum length the vine will grow.
4 Tips for Solving Function Questions on SAT Math
Now that you've seen the different kinds of function problems you may encounter on the SAT, let's go over some tips and strategies for solving different function problems.
#1: Use PIA and PIN
For many function equation problems, you can solve the problem more easily by plugging in answers (PIA) from the answer choices you're given. This is what we did in the function table example in the previous section. You can also use the technique of plugging in your own numbers (PIN) to test out points on function graphs or work with any variable function equation. Remember—most any time a problem has variables in the answer choices, you can use PIN.
#2: Remember to FOIL
The FOIL method (First, Outer, Inner, Last) can be very useful in solving function equations. For example, if you're asked to multiply $(x + 5)(2x -2)$, here's how you'd do it using FOIL.
First, multiply the first terms: $(x * 2x) =2x^2$
Next, multiply the outer terms: $(x * -2)= -2x$
Then, multiply the inner terms: $(5 * 2x)= 10x$
Finally, multiply the last terms: $(5 * -2) = -10$
Add those four values and simplify to get: $2x^2 + 8x -10$
Using FOIL makes it easier to solve functions SAT problems that require binomial multiplication, such as finding the product of two function expressions. It's a common error for students to see an equation like this: $(x + 3)^2$ and think the answer is $x^2 +9$. This is wrong! The correct answer, which you can use foil to get, is $x^2 + 6x + 9$.
#3: Start by Finding the y-intercept
Generally, the easiest place to begin when working with function graphs and tables is by finding the y-intercept. From there, you can often eliminate several different answer choices that don't match our graph or our equation.
The y-intercept is almost always the easiest piece to find, so it's always a good place to begin.
#4: Ordered Pairs to Test Your Equation
It's always smart to find two or more points (ordered pairs) of your functions and test them against a potential function equation. That's what we did for the function table question in the previous section. Sometimes one ordered pair works for your graph and a second does not.
You must match the equation to the graph (or the equation to the table) that works for every coordinate point/ordered pair, not just one or two.
Test Your Knowledge: Function Quiz
Now let's put your function knowledge to the test with a short quiz! Each of these five problems comes from SAT Practice Test 10, Math Module 1. Answers are at the bottom of this section, and you can find answer explanations here.
Question 1
Question 2
Question 3
Question 4
Question 5
Answers:
1: 77, 2: A, 3: D, 4: A, 5: A
Summary: Functions SAT Questions
Many students have not dealt a lot with functions, but don't let these kinds of questions intimidate or confuse you when you see them on the SAT. The principles behind functions are a simple matter of input, output, and plugging in values. Once you have those principles down, SAT function questions are pretty straightforward. If you can make it through the quiz in this article, you're in a great place to ace function questions on SAT Math!
What's Next?
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