Cracking the SAT Subject Test in Math 2, 2nd Edition (D) (E) RANGE AND DOMAIN IN GRAPHS The graph of a function gives important information about the function itself You can generally state a function[.]
(D) (E) RANGE AND DOMAIN IN GRAPHS The graph of a function gives important information about the function itself You can generally state a function’s domain and range accurately just by looking at its graph Even when the graph doesn’t give you enough information to state them exactly, it will often let you eliminate incorrect answers about the range and domain Take a look at the following graphs of functions and the information they provide: If you followed this line to the left, it would continue to rise forever Likewise, if you followed it to the right, it would continue to fall The range of this line (the set of y-values it occupies) goes on forever; the range is said to be “all real numbers.” Because the line also continues to the left and right forever, there are no x-values that the line does not pass through The domain of this function, like its range, is the set of all real numbers The same thing is true of all linear functions (whose graphs are straight lines); their ranges and domains include all real numbers There’s only one exception A horizontal line extends forever to the left and right (through all x-values) but has only one y-value Its domain is therefore all real numbers, while its range contains only one value The domain of this function is the set of all real numbers, because parabolas continue widening forever Its range, however, is limited The parabola extends upward forever, but never descends lower along the yaxis than y = −4 The range of this function is therefore {y: y ≥ −4} This function has two asymptotes Asymptotes are lines that the function approaches but never reaches They mark values in the domain or range at which the function does not exist or is undefined The asymptotes on this graph mean that it’s impossible for x to equal 2, and it’s impossible for y to equal 1 The domain of f(x) is therefore {x: x ≠ 2}, and the range is {y: y ≠ 1} The hole in this function’s graph means that there’s an x-value missing at that point The domain of any function whose graph sports a little hole like this one must exclude the corresponding x-value The domain of this function, for example, would simply be {x: x ≠ −2} To estimate range and domain based on a function’s graph, just use common sense and remember these rules: • If something about a function’s shape will prevent it from continuing forever up and down, then that function has a limited range • If the function has a horizontal asymptote at a certain y-value, then that value is excluded from the function’s range • If anything about a function’s shape will prevent it from continuing forever to the left and right, then that function has a limited domain • If a function has a vertical asymptote or hole at a certain x-value, then that value is excluded from the function’s domain • If you are asked to identify an asymptote, Plug In very large positive and negative numbers for x or y and see what values the other variable approaches Try 1, 1,000, −1, −1,000, etc • Sometimes you can Plug In The Answers (PITA) and see which values of x or y don’t make sense in the equation • Graphing the function on your calculator may be the easiest approach We’ll talk more about asymptotes in the next section Finding Asymptotes Without a Graph On the SAT Subject Test in Math you may be asked to find the asymptotes of a given function based on the expression of the function You can always graph such functions on your calculator and look for asymptotes, but it may be easier to know the rules for finding asymptotes and apply them to the given function Finding Vertical Asymptotes In order for a function to have asymptotes (vertical or horizontal), the function must be expressed as a fraction Finding a vertical asymptote is relatively straightforward Set the denominator of the function equal to 0 and solve for x There is a vertical asymptote at each value of x Furthermore, the domain of the function must exclude those values of x Finding Horizontal Asymptotes If the degree (largest exponent on x) of the numerator is equal to the degree of the denominator, there is a horizontal asymptote somewhere other than y = 0 (the x-axis) To find the horizontal asymptote, divide the coefficient of the leading term (the term with the highest exponent) of the numerator by the coefficient of the leading term of the denominator There will be a horizontal asymptote when y equals that value Furthermore, that value will be excluded from the range of the function Plugging In for Asymptotes Plugging In works well on these questions For instance you can Plug In The Answers on vertical asymptote questions by using the denominator of the fraction in the function If, when you Plug In, the denominator equals 0, then you have an asymptote at that value of x! On Horizontal Asymptote questions, Plug In a very large or very small number (like 100,000 or −100,000) for x The value of the function at that value of x should be very close to one of your answers; that answer is where an asymptote is If the degree of the denominator is greater than the degree of the numerator, there is a horizontal asymptote at y = 0 (the x-axis) If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote Instead, there will be a slant asymptote (which, luckily, is not tested on the SAT Subject Test in Math 2) Try an example: 32 Which of the following lines are asymptotes of ? I y = 3 II y = 0 III x = 1 (A) I only (B) II only (C) III only (D) I and III only (E) I, II, and III Here’s How to Crack It Start by looking for vertical asymptotes Set the denominator equal to zero and solve: x2 − 2x + 1 = 0 (x − 1)(x − 1) = 0 x = 1 Therefore, there is an asymptote at x = 1; eliminate (A) and (B) To find horizontal asymptotes, first look at the degrees of the numerator and denominator The degrees are equal, so there will not be an asymptote at y = 0; eliminate (E) Because the degrees are equal, take the leading terms (3x2 and x2) and divide the coefficients (remember, the coefficient of x2 is 1): = 3, so there is an asymptote at y = 3, so eliminate (C) and choose (D) DRILL 7: RANGE AND DOMAIN IN GRAPHS Test your understanding of range and domain with the following practice questions The answers can be found in Part IV 17 If the graph of y =f(x) is shown above, which of the following could be the domain of f ? (A) {x : x ≠ 0} (B) {x : x > 0} (C) {x : x ≥ 0} (D) {x : x > 1} (E) {x : x ≥ 1} 24 Which of the following could be the domain of the function graphed above? (A) {x : x ≠ 2} (B) {x : −2 < x 2} (D) {x : |x| ≠ 2} (E) {x : x > 2} 28 If y = g(x) is graphed above, which of the following sets could be the range of g(x) ? (A) {y : y ≤ −1} (B) {y : y ≥ −1} (C) {y : y ≥ −3} (D) {y : −3 ≤ y ≤ −1} (E) {y : y ≤ −3 or y ≥ −1} 37 Which of the following lines is an asymptote of the graph of y = 3e−2x + 5 ? (A) x = 0 (B) x = −2 (C) y = 5 (D) y = 0 (E) y = −6 ... other than y = 0 (the x-axis) To find the horizontal asymptote, divide the coefficient of the leading term (the term with the highest exponent) of the numerator by the coefficient of the leading... Plugging In works well on these questions For instance you can Plug In The Answers on vertical asymptote questions by using the denominator of the fraction in the function If, when you Plug In, the denominator equals 0,... denominator, there is no horizontal asymptote Instead, there will be a slant asymptote (which, luckily, is not tested on the SAT Subject Test in Math 2) Try an example: 32 Which of the following lines are asymptotes of