Cracking the SAT Subject Test in Math 2, 2nd Edition (D) 3x2 − 6 (E) 3(x − 2)2 DOMAIN AND RANGE Some function questions will ask you to make statements about the domain and range of functions With a f[.]
(D) 3x2 − 6 (E) 3(x − 2)2 DOMAIN AND RANGE Some function questions will ask you to make statements about the domain and range of functions With a few simple rules, it’s easy to figure out what limits there are on the domain or range of a function Domain The domain of a function is the set of values that may be put into a function without violating any laws of math When you’re dealing with a function in the f(x) form, the domain includes all of the allowable values of x Sometimes a function question will limit the function’s domain in some way, like the following: For all integers n, f(n) = (n − 2)π What is the value of f(7) ? Domain An easy way to think about it is that the domain is all the possible values of x In this function, the independent variable n is limited; n can be only an integer The domains of most functions, however, are not obviously limited Generally, you can put whatever number you want into a function; the domain of many functions is all real numbers Only certain functions have domains that are mathematically limited To figure out the limits of a function’s domain, you need to use a few basic rules Here are the laws that can limit a function’s domain Mathematical Impossibilities That Limit the Domain of a Function • Values that result in a denominator of zero: Any value that would make the denominator of a fraction equal zero must be excluded from the domain of the function • Any even root of a negative number (including square roots): An even root of a negative number will give an imaginary result Because domain requires real values for the function, any value that would result in taking the even root of a negative number must be excluded from the domain of the function Whenever a function contains a fraction, a square root, or another evennumbered root, it’s possible that the function will have a limited domain Look for any values that would make denominators zero, or evennumbered roots negative Those values must be eliminated from the domain Take a look at these examples In this function, there is a variable in the denominator of a fraction This denominator must not equal zero, so the domain of f(x) is {x ≠ 0} Once again, this function has a variable in the denominator of a fraction In this case, the value of x that would make the denominator equal zero is −5 Therefore, the domain of g(x) is {x ≠ −5} This function has a variable under a square root sign The quantity under a square root sign must not be negative, so the domain of t (a) is {a ≥ 0} Here again, you have a function with a variable under a square root This time, the values that would make the expression negative are values greater than 10; all of these values must be eliminated from the function’s domain The domain of s (a) is therefore {a ≤ 10} A function can involve both fractions and square roots Always pay careful attention to any part of a function that could place some limitation on the function’s domain It’s also possible to run into a function where it’s not easy to see what values violate the denominator rule or the square root rule Generally, factoring is the easiest way to make these relationships clearer For example: Here, you’ve got variables in the denominator You know this is something to watch out for, but it’s not obvious what values might make the denominator equal zero To make it clearer, factor the denominator Now, things are much clearer Whenever quantities are being multiplied, the entire product will equal zero if any one piece equals zero Any value that makes the denominator equal zero must be eliminated from the function’s domain In this case, the values 0, −4, and all make the denominator zero The domain of f(x) is {x≠ −4, 0, 2} Take a look at one more example Once again, you’ve got an obvious warning sign—variables under a radical Any values of x that make the expression under the radical negative must be eliminated from the domain But what values are those? Are there any? To make it clear, factor the expression The product of two expressions can be negative only when one of the expressions is negative and the other positive If both expressions are positive, their product is positive If both expressions are negative, their product is still positive So the domain of g(x) must contain only values that make (x + 5) and (x − 1) both negative, both positive, or one equal to zero The expression (x + 5) is zero at x = 5 and negative when x < −5 The expression (x − 1) is zero at x = 1 and negative when x < Between −5 and 1, (x − 1) is negative and (x + 5) is positive Therefore, the product of the two expressions will be negative when −5 < x < 1; this must be excluded from the domain of the function All other real values of x are in the domain; therefore, the domain of g(x) is {x ≤ −5} or {x ≥ 1} Domain Notation The domain of a function is generally described using the variable x A function f(x) whose domain includes only values greater than 0 and less than 24, could be described in the following ways: The domain of f(x) is {0 < x < 24} The domain of f is the set {x: 0 < x < 24} A function in the form f(x) can be referred to either as f(x) or simply as f Range The range of a function is the set of possible values that can be produced by the function When you’re dealing with a function in the f(x) form, the range consists of all the allowable values of f(x) The range of a function, like the domain, is limited by a few laws of mathematics Several of these laws are the same laws that limit the domain Here are the major rules that limit a function’s range • An even exponent produces only nonnegative numbers Any term raised to an even exponent must be positive or zero • The square root of a quantity represents only the positive root Like even powers, a square root can’t be negative The same is true for other even-numbered roots ( , , etc.) • Absolute values produce only nonnegative values Range An easy way to think about it is that the range is all possible values of y In the case of functions, the range is all the possible values of f(x) These three operations—even exponents, even roots, and absolute values —can produce only nonnegative values Consider these three functions These functions all have the same range, {f(x) ≥ 0} These are the three major mathematical operations that often limit the ranges of functions They can operate in unusual ways The fact that a term in a function must be nonnegative can affect the entire function in different ways Take a look at the following examples Each of these functions once again contains a nonnegative operation, but in each case the sign is now flipped by a negative sign The range of each function is now {f(x) ≤ 0} In addition to being flipped by negative signs, ranges can also be slid upward or downward by addition and subtraction Take a look at these examples Each of these functions contains a nonnegative operation that is then decreased by 5 The range of each function is consequently also decreased by 5, becoming {f(x) ≥ −5} Notice the pattern: A nonnegative operation has a range of {f(x) ≥ 0} When the sign of the nonnegative operation is flipped, the sign of the range also flips When a quantity is added to the operation, the same quantity is added to the range These changes can also be made in combination In this function, the sign of the nonnegative operation is flipped, is added, and the whole thing is divided by 2 As a result, the range of g(x) is {g(x) ≤ 3} The range of x2, which is {y: y ≥ 0}, has its sign flipped, is increased by 6, and is then divided by 2 Range Notation Ranges can be represented in several ways If the function f(x) can produce values between −10 and 10, then a description of its range could look like any of the following: • The range of f(x) is given by {f: −10 < f(x) < 10} • The range of f(x) is {−10 < f(x) < 10} • The range of f(x) is the set {y: −10 < y < 10} Range and POE As with many problems on the SAT Subject Test in Math 2, range problems are often best handled using process of elimination Here are a few tips to help you eliminate answer choices on these problems: • Look for negative values Remember that even exponents, even roots, and absolute value must always be nonnegative • Similarly, a negative in front of any of the above makes that part of the expression always not positive • Plugging In often works great! Plug In a value for x, find f(x), and eliminate any answer that doesn’t include the result in its range Solving a Range Question Now that you’ve learned about ranges, let’s try out a question Take a look at the following example 13 If f(x) = |−x2 −8| for all real numbers x, then which of the following sets is the range of f? (A) {y: y ≥ −8} (B) {y: y > 0} (C) {y: y ≥ 0} (D) {y: y ≤ 8} (E) {y: y ≥ 8} Here’s How to Crack It Start out with what you know about the equation Since the result of absolute value is a nonnegative number, you can eliminate (A) right away Is there a maximum number that an absolute value creates? No So you can also eliminate (D) Now look at x2 We know that there’s no maximum that x2 can be, but there is a minimum The smallest x2 can be is 0 If x2 = 0, then the result inside the absolute value sign would be −8 This means that, when x = 0, f(x) = 8 So the answer is (E) Now you may be thinking, but what about that negative sign? Well, increasing the value of x would make −x2 smaller, and that smaller number minus would be even smaller Then the absolute value would make all that negativity positive, which confirms that the least value of the function is 8 Plugging In on Range Questions Because all questions on the SAT Subject Test in Math 2 are multiple choice, you can always Plug In and use POE on range questions It may take a little longer but it gives you a chance to score another point So, if you’re confused by the process of finding the range, or not sure what steps to take on a particular range question, Plug In! Let’s take another look at question 13 on the previous page If you Plugged In x = 3, you would find that f(3) = 17 From that info you could eliminate (D) If you Plugged In 0, you’d see that f(0) = If you Plugged In numbers less than 0, you’d see that f(x) never gets smaller than 8 The answer is (E) You still get to the right answer! Using Your Calculator to Solve a Range Question You can also use your calculator to help you solve range questions Let’s take a look: 49 If , then what is the range of f(x)? (A) {y: y ≥ −7.625} (B) {y: y ≥ −7} (C) {y: y ≥ −6.375} (D) {y: y ≤ −7.625} (E) {y: y ≥ 0} Here’s How to Crack It: Without your calculator, you need to figure out if the quadratic in the numerator of the fraction has real roots (which would make the expression within the absolute value bracket potentially equal to zero) and go from there With a graphing calculator, you can use the “y =” function instead On the TI-84, go to the “y =” menu and input the function (note: absolute value can be found under MATH->NUM->abs) Be careful with parentheses, especially with the fraction and the absolute value! ... must be excluded from the domain of the function All other real values of x are in the domain; therefore, the domain of g(x) is {x ≤ −5} or {x ≥ 1} Domain Notation The domain of a function is generally... must be eliminated from the domain Take a look at these examples In this function, there is a variable in the denominator of a fraction This denominator must not equal zero, so the domain of f(x) is {x ≠ 0}... denominator must not equal zero, so the domain of f(x) is {x ≠ 0} Once again, this function has a variable in the denominator of a fraction In this case, the value of x that would make the denominator equal zero is −5 Therefore, the domain of g(x) is {x ≠ −5}