Evaluating Functions
Evaluating Functions
Evaluating a function simply means finding f(x) at some specific value x. To put it more bluntly, these are glorified substitution questions. We glorify them above all because they’re easy. Here’s an example:
If f(x) = x2 – 3, what is f(5)?
See how that f(5) substituted a 5 for the x in f(x)? Well, every time you see an x in the equation, replace it with a 5:
You almost don’t even have to think at all when answering these questions. If the entire Math section was just a bunch of evaluating functions questions, amoebas could get 800s and Ivy League schools would welcome every well-rounded single-celled organism who applied.
Ah, but life and the new SAT ain’t that easy. Here’s one wrinkle the new test may throw at you. You may have to evaluate a function at a variable rather than a constant. For example,
If , what is f(x + 1)?
Okay, slightly harder than substituting in a number, but still not difficult. Search out all the occurrences of x in the function and replace it with (x + 1):
As long as you remembered to distribute that negative sign across the (x + 1) to make x – 1 in that second step, you’re all set.
Performing Operations on Functions
Functions can be added, subtracted, multiplied, and divided like any other quantities. A few key rules will make these operations easier. For any two functions f(x) and g(x),
Rule Example
Addition If f(x) = x2 and g(x) = 2x:
(f + g)(x) = x2 + 2x
Subtraction If f(x) = x2 + 5 and g(x) = x2 + 2x + 1:
(fg)(x) = x2 + 5 – x2 – 2x – 1 = –2x + 4
Multiplication If f(x) = x and g(x) = x3 + 8:
Division If f(x) = 2x and g(x) = x2:

Here’s a quick rule to follow for all of these operations on functions: Work out the value for both functions separately, and then perform the operation on those two values. Remember that any time you divide functions,
the resulting function is undefined whenever the g(x) in the denominator equals zero. Division by zero is always a no-no.
Wacko Symbols Questions (Algebra in Disguise)
The SAT seems to give itself a cooky thrill by creating odd symbols and then defining those symbols as mathematical functions. For example, a typical symbol SAT question might say,
Let a @ b be defined as , where What is the value of 4 @ 2?
These symbols questions are just snazzy, dressed-to-kill, evaluating functions questions. Answer them by plugging in:
Some students get frazzled when they see odd symbols in their test booklet, which is exactly what the SAT wants. Don’t get tripped up on these otherwise easy questions.
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