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Consider these two practice GMAT Quantitative problems:<\/p>\nConsider these two practice GMAT Quantitative problems:<\/p>\n
1) Given f(x) = 3x \u2013 5, for what value of x does 2*[f(x)] \u2013 1 = f(3x \u2013 6)<\/p>\n
If you find these questions completely incomprehensible, then you have found the right blog post.<\/p>\n <\/p>\n The GMAT Quantitative section will ask an occasional question about function notation.\u00a0 Here is a basic catechism about functions and what you need to know about them for the GMAT.<\/p>\n <\/p>\n A function is a rule, a \u201cmachine\u201d, that takes an input and gives an output.\u00a0 When we are told the equation of a function, that equation makes explicit the rule this particular function is following.\u00a0 For example, for the function f(x) = 3x \u2013 5, the rule is: whatever input x you give ---- and that input could be any real number ---- this function will multiply this input by 3 and then subtract five from the product: that difference is the output.\u00a0 If I put in an input of x = 2, then I get an output of 3(2) \u2013 5 = 1, and the way we compactly write that fact with function notation is: f(2) = 1.\u00a0 In other words, an input of 2 gives an output of 1.<\/p>\n Notice --- this is a very subtle issue.\u00a0 The x that appears in the equation of a function is a different sort of variable than the ordinary x of solve-for-x algebra.\u00a0 This x is what one might call a \u201cformula variable\u201d, like the a, b, and c in the quadratic formula.\u00a0 In other words, the x of function notation is not an x that is equal to only a single value; rather, it can be set equal to any value, any real number on the number line, when we want to plug that number into the function.<\/p>\n <\/p>\n When we write f(x), many people new to function notation will misinterpret this as multiplication --- as if there\u2019s a thing \u201cf\u201d times the variable \u201cx\u201d.\u00a0 That is 100% incorrect.\u00a0 A function is a process through which the input goes.\u00a0 Cooking is a process through which food goes.\u00a0 Puberty is a process through which people go.\u00a0 A function is a process acting on the number, and the nature of that action is outside the categories of simple arithmetic actions (add, subtract, multiply, divide).<\/p>\n Relatedly, the parentheses of function notation are mathematically inviolable.\u00a0 Nothing may pass through these parentheses.\u00a0 Again, this can be anti-intuitive, because when parentheses are used in ordinary notation, you can distribute through parenthesis, factor out, etc. \u00a0Because a function is a different category of mathematical object, its parentheses are of a different nature.\u00a0 Thus<\/p>\n If you can simply avoid these mistakes and respect at all times the inviolability of the function\u2019s parentheses, you will already be in better shape than a sizable portion of GMAT test takers.<\/p>\n <\/p>\n In the above section, I discussed ways that folks new to functions might misinterpret function notation.\u00a0 Now, I am going to discuss how functions are seen by people who really understand them.\u00a0 Suppose we have the function f(x) = 3x \u2013 5.\u00a0 Here\u2019s what a mathematician looking at this function sees:<\/p>\n Where folks new to function just see the letter x, mathematicians see a \u201cbox\u201d, an empty slot, a space that is, in some ways, analogous to an artist\u2019s blank canvas.\u00a0 Anything that get plugged into the box on the left needs to get plugged into the box on the right.\u00a0 We can plug in numbers --- any of the continuous infinity of real numbers on the real number line.\u00a0 We can also plug in algebraic expressions: If I put (2x + 7) into the box on the left, I need to put that exact identical expression into the box on the right.\u00a0 I can even put whole functions --- the same function or an entirely different function --- into the boxes.\u00a0 In fact, the list of mathematical objects that can be plugged into a function extends into far more sophisticated mathematical objects (matrices, differential operators, etc.) that are well beyond the realm of GMAT Quant.\u00a0 The GMAT, though, will expect you to know what to do if they give you, say, the function f(x) = 3x \u2013 5, and then ask you, say, to plug in the expression 2x + 7:<\/p>\n f(2x + 7) = 3*(2x + 7) \u2013 5 = 6x + 21 \u2013 5 = 6x + 16<\/p>\n <\/p>\n If this is your first time encountering, or first time understanding, function notation, it is a worthwhile topic to practice, so that you are comfortable with it by test day.\u00a0 If you feel you have learned something from this, go back and try those two practice problems again before reading the solutions below.\u00a0 Also, here\u2019s a practice question from our product:<\/p>\n 3)\u00a0https:\/\/gmat.magoosh.com\/questions\/147<\/a><\/p>\n <\/p>\n 1) We have the function f(x) = 3x \u2013 5, and we want to some sophisticated algebra with it.\u00a0 Let\u2019s look at the two sides of the prompt equation separately.\u00a0 The left side says: 2*[f(x)] \u2013 1 ---- this is saying: take f(x), which is equal to its equation, and multiply that by 2 and then subtract 1.<\/p>\n 2*[f(x)] \u2013 1 = \u00a02*(3x \u2013 5) \u2013 1 \u00a0= 6x \u2013 10 \u2013 1 = 6x \u2013 11<\/p>\n The right side says f(3x \u2013 6) --- this means, take the algebraic expression (3x \u2013 6) and plug it into the function, as discussed above in the section \u201cHow a mathematician things about a function.\u201d\u00a0 This algebraic expression, (3x \u2013 6), must take the place of x on both sides of the function equation.<\/p>\n f(3x \u2013 6)= 3*[3x \u2013 6] \u2013 5 = 9x \u2013 18 \u2013 5 =\u00a0 9x \u2013 23<\/p>\n Now, set those two equal and solve for x:<\/p>\n 9x \u2013 23 = 6x \u2013 11<\/p>\n 9x = 6x \u2013 11 + 23<\/p>\n 9x = 6x + 12<\/p>\n 9x \u2013 6x = 12<\/p>\n 3x = 12<\/p>\n x = 4<\/p>\n Answer =\u00a0B<\/strong><\/p>\n <\/p>\n 2) There are several ways to approach this problem.\u00a0 One quick way is to notice that if x = 2, f(2) = 2\/3.\u00a0 That\u2019s not the answer, but it gives us a shortcut.\u00a0 If f(k) = 2, then we see that f(f(k)) = f(2) = 2\/3.\u00a0 So, really, finding the value of k that satisfies the prompt equation really simplifies to solving the equation f(k) = 2.<\/p>\n f(k) = k\/(k + 1) = 2<\/p>\n Multiply both sides by the denominator.<\/p>\n k = 2*(k + 1)<\/p>\n k = 2k + 2<\/p>\n k \u2013 2k = 2<\/p>\n \u2013k = 2<\/p>\n Multiply both sides by \u20131.<\/p>\n k = \u20132<\/p>\n answer =\u00a0A<\/strong><\/p>\n This post was written by Mike McGarry, GMAT expert at\u00a0Magoosh<\/a>, and originally posted\u00a0here<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":" Consider these two practice GMAT Quantitative problems: 1) Given f(x) = 3x \u2013 5, for what value of x does 2*[f(x)] \u2013 1 = f(3x \u2013 6) 0 4 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<\/a><\/p>\n\n
Function notation<\/h2>\n
What is a function?<\/h2>\n
Typical misunderstandings of function notation<\/h2>\n
<\/a><\/p>\nHow a mathematician thinks about a function<\/h2>\n
<\/a><\/p>\nSummary<\/h2>\n
Practice problem explanations<\/h2>\n