Determine which of the following relations represent y as a function of x. Find the domainand range of those relations which are functions.(a) {(−3, 9), (−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4), (3, 9)}(b) {(−3, 0), (1, 6), (2, −3), (4, 2), (−5, 6), (4, −9), (6, 2)}(c) {(−3, 0), (−7, 6), (5, 5), (6, 4), (4, 9), (3, 0)}(d) {(1, 2), (4, 4), (9, 6), (16, 8), (25, 10), (36, 12), . . .}(e) {(x, y) : x is an odd integer, and y is an even integer}(f) {(x, 1) : x is an irrational number}(g) {(1, 0), (2, 1), (4, 2), (8, 3), (16, 4), (32, 5), . . . }(h) {. . . , (−3, 9), (−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4), (3, 9), . . . }(i) {(−2, y) : −3 < y < 4}(j) {(x, 3) : −2 ≤ x < 4}
Question
Determine which of the following relations represent y as a function of x. Find the domainand range of those relations which are functions.(a) {(−3, 9), (−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4), (3, 9)}(b) {(−3, 0), (1, 6), (2, −3), (4, 2), (−5, 6), (4, −9), (6, 2)}(c) {(−3, 0), (−7, 6), (5, 5), (6, 4), (4, 9), (3, 0)}(d) {(1, 2), (4, 4), (9, 6), (16, 8), (25, 10), (36, 12), . . .}(e) {(x, y) : x is an odd integer, and y is an even integer}(f) {(x, 1) : x is an irrational number}(g) {(1, 0), (2, 1), (4, 2), (8, 3), (16, 4), (32, 5), . . . }(h) {. . . , (−3, 9), (−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4), (3, 9), . . . }(i) {(−2, y) : −3 < y < 4}(j) {(x, 3) : −2 ≤ x < 4}
Solution
Sure, let's go through these one by one:
(a) This is a function. Each x value is paired with exactly one y value. The domain is {-3, -2, -1, 0, 1, 2, 3} and the range is {0, 1, 4, 9}.
(b) This is not a function. The x value 4 is paired with two different y values (2 and -9).
(c) This is a function. Each x value is paired with exactly one y value. The domain is {-7, -3, 3, 4, 5, 6} and the range is {0, 4, 5, 6, 9}.
(d) This is a function. Each x value is paired with exactly one y value. The domain is {1, 4, 9, 16, 25, 36, ...} and the range is {2, 4, 6, 8, 10, 12, ...}.
(e) This is a function. Each x value (odd integer) is paired with exactly one y value (even integer). The domain is all odd integers and the range is all even integers.
(f) This is a function. Each x value (irrational number) is paired with exactly one y value (1). The domain is all irrational numbers and the range is {1}.
(g) This is a function. Each x value is paired with exactly one y value. The domain is {1, 2, 4, 8, 16, 32, ...} and the range is {0, 1, 2, 3, 4, 5, ...}.
(h) This is a function. Each x value is paired with exactly one y value. The domain is all integers and the range is {0, 1, 4, 9}.
(i) This is not a function. The x value -2 is paired with multiple y values between -3 and 4.
(j) This is a function. Each x value between -2 and 4 is paired with exactly one y value (3). The domain is all x such that -2 ≤ x < 4 and the range is {3}.
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