Compound Functions
Compound Functions
You know those Russian nesting dolls? Each doll has a smaller and smaller doll inside it? Compound functions are like that. A compound function is a function that operates on another function. It’s written out like this: f(g(x)). To evaluate a compound function, first evaluate the internal function, g(x). Next, evaluate the outer function at the result of g(x). It’s just double substitution: a classic SAT question that looks much meaner than it really is.
Try this example on for size:
Suppose h(x) = x2 + 2x and j(x) = | + 2|. What is j(h(4))?
First evaluate h(4):
Now plug 24 into the definition of j:
Just make sure you pay attention to the order in which you evaluate the compound function. Always evaluate the inner function first. If the question had asked you to evaluate h(j(4)), you’d get a completely different answer:
As with ordinary evaluating functions questions, the SAT doesn’t always give you a constant with which to evaluate compound functions.
Suppose f(x) = 3x + 1 and g(x) = . What is g(f(x))?
When you aren’t given a constant, just substitute the definition of f(x) as the input to g(x). It’s as if you’re being asked to evaluate a signle function at a variable rather than a constant.
Compound Wacko Symbols Questions
The SAT also sometimes asks compound symbols questions. These are exactly the same as compound function questions.
Let a#b#c#d be defined for all numbers by a#b#c#d = ab – cd. If x = 6#3#5#4, then what is the value of 7#x#3#11?
Strange symbols are flying all over the place, and the question is asking you to calculate the value of a strange symbol with a variable in it?! No problem. The answer to this question is only two steps away:
  1. Calculate the value of x.
  2. Calculate the value of 7#x#3#11 (which won’t be very hard, since by step 2, you’ll know exactly what x equals).
Since
Now plug x = –2 into 7#x#3#11:
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