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Cracking the SAT Subject Test in Math 2, 2nd Edition COMPOUND FUNCTIONS Compound functions combine functions by using the output of one function as the input of the other The first thing to remember a[.]
COMPOUND FUNCTIONS Compound functions combine functions by using the output of one function as the input of the other The first thing to remember about compound functions is order of operations: you what’s in the parentheses first For example: As you can see, for most compound functions, the order in which the functions apply matters greatly in what the result will be! f ∘ g? Sometimes ETS will use the notation f ∘ g This notation means that the second function is inside the parentheses; in other words, f ∘ g= f(g(x)) In either case, you work from right to left Luckily, on the SAT Subject Test in Math 2, there are many techniques that can greatly simplify compound function questions In particular, Plugging In works very well on these types of questions Read on to see examples of different ways ETS will use compound functions on this test Plugging In Values Given by ETS The simplest compound function questions on the SAT Subject Test in Math 2 are the questions in which you are given the number to plug in to the function These questions often have wrong answers that misuse negative numbers or violate order of operations For example: The expression g(f(x)) is a compound function made up of the functions f(x) and g(x) As with any algebraic expression with parentheses, you start with the innermost part To find g(f(x)) for any x, calculate the value of f(x), and plug that value into g(x) The result is g(f(x)) Like questions based on simple algebraic functions, compound-function questions come in two flavors—questions that require you to plug numbers into compound functions and the arithmetic, and questions that require you to plug terms with variables into compound functions and find an algebraic answer For example: 16 What is the value of g(f(−4)) ? (A) 0.11 (B) 1.00 (C) 2.75 (D) 5.41 (E) 6.56 Here’s How to Crack It To find the value of g (f(−4)), just plug −4 into f(x); you should find that f(−4) = −21 Then, plug −21 into g(x) You should find that g(−21) = The correct answer is (B) The other answers in this problem illustrate how ETS anticipates where you can mess up For example, if you use 4 instead of −4 in f(x), you find f(4) = 59, and g(59) = 0.11, which is (A) If, however, you were to reverse order of operations and find f(g(−4)), you would find an answer of 5.41, which is (D) PITA and Compound Functions Another relatively simple type of compound function question asks you to find the inside function when you’re given the outside function and a value for the compound function Despite the use of compound function notation, these questions are really the same as normal function notation questions; they just look more complicated In these cases, Plugging In the Answers is the best approach 12 If f(x) = 4x − 2 and f(g(x)) = 10, then g(x) = (A) (B) (C) 10 (D) 4x + 2 (E) Here’s How to Crack It This question is looking for a value that, when put into f(x), gives a result of 10 The question just happens to call that “something” g(x) You know that one of the five answer choices must work, so Plug In the Answers Choices (D) and (E) contain variables; therefore, plugging those answers into f(x) will result in an answer choice with variables You want f(g(x)) = 10, with no variables, so eliminate (D) and (E) Choice (B) is the middle choice of those that remain, so plug 3 into f(x): f(3) = 4(3) − 2 = 10 This matches what the question says, so choose (B) Finding the Algebraic Expression of a Compound Function A more complicated way that ETS tests compound functions is by asking for the algebraic expression of a compound function Luckily, the SAT Subject Test in Math 2 is a multiple-choice test, so you don’t have to come up with the expression yourself Instead, you can (and should) Plug In a value for x and turn these questions into ones like question 12 above Try an example f(x) = x2 + 10x + 3 21 Which of the following is g(f(x)) ? (A) (B) (C) (D) (E) Here’s How to Crack It Instead of doing lots of messy algebra, just pick an easy number to Plug In for x Let’s try x = 3 So you’re looking for g(f(3)) Work from the inside out, f(3) = 42, so g(f(3)) = g(42) When you plug 42 into g, you get , the target number Plugging x = 3 into the answer choices, you find that (B) hits that target Finding an Original Function When Given the Compound Function A variation on asking for the algebraic function of a compound function is to ask for one of the original functions given the other original function and the compound function As above, the best way to approach these questions is to Plug In However, you still need to respect order of operations and Plug In accordingly For example: 50 If f(x) = (x + 2)2 + 3 and , then g(x) = (A) x + 3 (B) (C) (x + 2)2 − 3 (D) (E) Here’s How to Crack It Start by Plugging In to the compound function Make x = 2, so Next, plug x = 2 into the answer choices to find the possible values for g(2): (A) (2) + 3 = 5 (B) (C) ((2) + 2)2 = −3 = 13 (D) (E) Using “y =” and “CALC->value” on function questions For some function questions, you will find yourself trying many values of x in a given function On the TI-84, you can save yourself a lot of work by using the “y =” and “value” functions on your calculator First, press the “y =” button and input the function Be careful with parentheses, especially with functions using fractions! Next, press “2ND” “TRACE” to access the “CALC” menu The first option is “value”; select that option and input the x-value you are looking for Your calculator will give you the value of f(x) at that spot; type in another number to find another f(x) If you get an error, simply resize the window using the “WINDOW” tool so you can see the point you’re requesting Remember that you’re looking for the g(2) that, put into f(x), gives you the value of f(g(2)) you just found Therefore, plug each value for g(2) into the original f(x), and find the one that equals 7.84: (A) f(5) = (5+2)2 + 3 = 52 (B) f(0.2) = (0.2 + 2)2 + 3 = 7.84 (C) f(13) = (13 + 2)2 + 3 = 228 (D) f(1.44) = (1.44 + 2)2 + 3 = 14.8336 (E) f(7.84) = (7.84 + 2)2 + 3 = 99.8256 The answer is (B) DRILL 3: COMPOUND FUNCTIONS Practice working with compound functions in the following questions The answers can be found in Part IV If f(x) = 3x and g(x) = x + 4, what is the difference between f(g(x)) and g(f(x)) ? (A) (B) (C) (D) (E) 12 If f(x) = |x| − 5 and g(x) = x3 − 5, what is f(g(−2)) ? (A) −18 (B) −5 (C) (D) (E) If f(x) = 5 + 3x and f(g(x)) = 17, then g(x) = (A) (B) (C) 56 (D) 3 + 5x (E) 5 + 3x 16 Which of the following is g(f(x)) ? (A) x − 1 (B) x + 1 (C) x + 7 (D) x + 9 (E) x2 − 2x − 1 20 What is the positive difference between f(g(3)) and g(f(3)) ? (A) 0.7 (B) 0.9 (C) 1.8 (D) 3.4 (E) 6.8 22 If and f(g(x)) = 2, then g(x) = (A) 0.01 (B) 10 (C) 20 (D) 100 (E) 200 35 If f(x) =x2 − 1 and (A) (B) (C) , then g(x) = ... Work from the inside out, f(3) = 42, so g(f(3)) = g(42) When you plug 42 into g, you get , the target number Plugging x = 3 into the answer choices, you find that (B) hits that target Finding an Original Function When Given the. .. To find the value of g (f(−4)), just plug −4 into f(x); you should find that f(−4) = −21 Then, plug −21 into g(x) You should find that g(−21) = The correct answer is (B) The other answers in this problem illustrate how ETS anticipates where... simply resize the window using the “WINDOW” tool so you can see the point you’re requesting Remember that you’re looking for the g(2) that, put into f(x), gives you the value of f(g(2)) you just found Therefore,