Algebra
Functions
The new SAT has changed to the way functions look. But the way functions work remains the same. Some makeover!
Functions are like meat grinders, except that you use numbers instead of animal flesh. With every function, you start with an initial number. The function grinds it—that is, performs some mathematical operation—then spits out a finished number. Here’s a simple example:
If f(x) = 4x – 6, what is the value of f(7)?
The term f(x) is what’s used to denote a function, and the 4x – 6 shows what operation you’ll perform when you get an input number. Once you’re given the 7, start grinding:
f(x) = 4x – 6
f(7) = 4(7) – 6
f(7) = 28 – 6
f(7) = 22
There it is. You put in a 7, and the function spits out a value of 22.
Those are the basics. That’s a simple function, and as you should know by now, when you get to the tougher items on the SAT, nothing is that simple. There are four basic ways to make a function item more difficult:
  1. Using a strange mathematical symbols.
  2. Using a compound function.
  3. Placing the function on a coordinate grid.
  4. Talking about the domain and range of a function.
Strange Math Symbols
There’s nothing fancy about the first option, except that the SAT will use a symbol such as @ instead of f to signify a function. So don’t be alarmed if you see @x: it’s the same exact thing as f(x).
Compound Functions
Think of a compound function as a pregnant function. If f(x) were carrying the baby function g(x), the compound function would look like f(g(x)). What this means is that you have to grind the numbers for g(x)—the baby comes first—and once you get an output, you place that number into the f(x) function. Then you can hand out cigars.
9. If and , then what is the value of t(r(16))?
Nice and confusing—just the way the test-makers like it. But also quite solvable. Everything’s given to you in the item. All you have to do is follow the directions.
Because the r function is inside the t function, you have to run that function first:
Now that you have that value, place it into the t function. You can also do a little factoring beforehand to get rid of the denominator:
Your answer is . You can either grid in 81⁄16 or convert it to the decimal approximation 5.06, but that’s no huge hurdle for you to jump.
Functions and Coordinate Grids
You are probably familiar with a coordinate grid, but let’s look at one below anyway:
The horizontal line is the x-axis, and the vertical one is the y-axis. The place where they meet is the origin. Using this system, you can place any point on the grid if you give it an x value and a y value, conventionally written as (x, y). Point A is at (7, 2), and point B is at (–4, –1). The positive and negative values come up because every x value right of the y-axis is positive, and everything left of it is negative. For y values, all points above the x-axis are positive, and all values below it are negative.
That’s Coordinate Grid 101 for you. But you can graph more than just points on a grid. If you have two or more points, you have a line.
Now think about a basic function. You put in an initial value called x, and you get an f(x) value. If you call f(x) the y value, you can see how a typical function can spit out a huge number of points that can then be graphed.
Look back to our simple function f(x) = 4x – 6. If you put in x = 0, you get:
f(x) = 4x –6
f(0) = 4(0) – 6
f(0) = –6
So (0, –6) is one point you can graph from this function. It’s there at point C. If you place x = 1 into the function, you get:
f(x) = 4x – 6
f(1) = 4(1) – 6
f(1) = –2
That’s point D at (1, –2). You can keep plugging in values and get points like E and F, but there’s no need. Once you have two points spit out by the function, you can draw a line connecting them. You have now graphed a function.
Some functions will be easier to graph than others. Because f(x) = 4x – 6 has no squared terms, it’s going to be only a line. More complicated functions might have more complicated graphs, but the basic point is still the same: stick in an x value, work the math and let the function churn out a corresponding y value, then graph that point at (x, y).
Domain and Range
From the previous example, it would seem that functions are interchangeable with equations, but there is a key difference: for every value of x placed in the function, there can be only one value for f(x). You can see this easily on a graph.
Imagine you’re a Samurai warrior. (If you actually are a Samurai warrior, this shouldn’t be too hard.) Whip out your function-testing kitana—that’s your sword, Samurai—and slice down vertically on the graph. If your blade passes through a line or figure only once, no matter where you slice down vertically, that means there is only one value of f(x) for every input of x. You can try it out on the Good Snake and see this is true. On the Bad Snake, your blade passes through the figure at multiple points. This evil snake is not a function.
Getting back to domain and range, the domain is the set of all the x values a function can have. The range is the total set of f(x) values that can be generated by a function. Sometimes there are restrictions on the domain or range of a function. These are certain values a function just can’t have. Consider the function . If x is any number less than 11, you’ll wind up with a negative value under the square root sign. That’s bad. Really bad. So the domain of this function is restricted in the sense that x must be greater than 11, or x > 11. But wait, there’s more excitement to come. Looking at the denominator of the fraction, you can see that if x = –3, you would have a zero in the denominator, meaning the fraction would be undefined. That’s bad. Really bad. So a further restriction is that x can’t equal –3, or .
Restrictions on range work the same way. There will be some functions that limit the value f(x) might have. The simplest example is . No matter what value for x you place in there, your f(x) will be positive because it will be multiplied by itself. So the range of this function is restricted to all positive numbers.
The domain and range of functions are lesser known terms meant to scare you. If you know what they mean, you should be able to determine the answer to an item that asks for them.
Functions As Models
Try this in some open space, like a park:
  • Run as fast as you can for 15 seconds.
  • Sit down for the next 15 seconds.
  • Now crawl for 15 seconds.
  • Now run back toward your starting point for 15 seconds.
What have you accomplished? For starters, you should have convinced anyone watching you that you’re a bit touched in the head. You have also completed a real-life situation that can be graphed. The “model” of your activity would look like this:
You can see how the model accurately shows each different activity. Because you were running in the first 15 seconds, you covered a lot of distance. Then no distance was covered while you sat on your rump. Crawling moved you forward but not as much as running because, well, you were crawling. Then you ran backward, decreasing your distance from the starting point.
A function as model item will pose a similar question. You might be given a graph and asked which set of activities that graph matches. Or you might be given a set of activities and asked which graph correctly models the activity. Either way, it’s a matter of looking at the activity and translating it into graph form.
That covers the algebra basics. Now let’s apply these concepts to some SAT strategies.
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