This question tests understanding of the function definition: each input has exactly one output (one y per x), not allowing one input with multiple outputs. A function assigns exactly one output to each input: {(1,3),(2,5),(3,7)} is a function (x=1→3, x=2→5, x=3→7, each input once with one output), but {(2,3),(2,5),(4,7)} is NOT (x=2 maps to both 3 and 5, violates rule—input 2 has two outputs); graphically, the vertical line test applies (if any vertical line hits the graph more than once, it's not a function—that x has multiple y values); equations like y=2x+1 are functions (each x input gives exactly one y output by computation). In this graph with points (0,2), (0,-2), (1,1), and (-1,1), input x=0 maps to both 2 and -2, which is a violation, and the vertical line at x=0 intersects twice. This relation is not a function based on the one-output rule because input 0 does not have exactly one output. A common error is applying the vertical line test horizontally (wrong axis) or thinking no repeated y-values are needed (but here the issue is multiple y for same x). To check relations like this, (1) identify inputs (x-values or domain), (2) check each input (does it map to one output or multiple?), and (4) for graphs, use the vertical line test (imagine a vertical line sliding across; it hits twice at x=0). Common mistakes include incorrectly applying the vertical test as horizontal (wrong axis) or confusing 'same y for different x' as a violation (that's okay—many-to-one allowed, but not the case here).

This question tests understanding of the function definition: each input has exactly one output (one y per x), not allowing one input with multiple outputs. A function assigns exactly one output to each input: {(1,3),(2,5),(3,7)} is a function (x=1→3, x=2→5, x=3→7, each input once with one output), but {(2,3),(2,5),(4,7)} is NOT (x=2 maps to both 3 and 5, violates rule—input 2 has two outputs); graphically, the vertical line test applies (if any vertical line hits the graph more than once, it's not a function—that x has multiple y values); equations like y=2x+1 are functions (each x input gives exactly one y output by computation). In this set of ordered pairs {(1,6),(2,12),(3,18),(4,24)}, the inputs are 1, 2, 3, and 4, each appearing once and mapping to a single output: 1→6, 2→12, 3→18, 4→24, with no input having multiple outputs. This relation is a function based on the one-output rule because every input has exactly one corresponding output, and there are no duplicates in the x-values with differing y-values. A common error is thinking repeated y-values violate the function rule (wrong—different x can share y, but here y-values are all different anyway), or mistakenly believing functions require the same output for all inputs. To check relations like this, (1) identify inputs (x-values or domain), (2) check each input (does it map to one output or multiple?), and (5) for ordered pairs, look for the same x with different y (which would be a violation). Common mistakes include confusing 'same y for different x' as a violation (that's okay—many-to-one allowed) or claiming equations always functions (implicit relations like x²+y²=1 aren't—solving for y gives ±√(1-x²), two outputs).

This question tests understanding of the function definition: each input has exactly one output (one y per x), not allowing one input with multiple outputs. A function assigns exactly one output to each input: {(1,3),(2,5),(3,7)} is a function (x=1→3, x=2→5, x=3→7, each input once with one output), but {(2,3),(2,5),(4,7)} is NOT (x=2 maps to both 3 and 5, violates rule—input 2 has two outputs); graphically, the vertical line test applies (if any vertical line hits the graph more than once, it's not a function—that x has multiple y values); equations like y=2x+1 are functions (each x input gives exactly one y output by computation). This question asks for the core property, which is that each input has exactly one output, distinguishing functions from general relations where inputs might have multiple or no outputs. The correct statement aligns with the one-output rule, emphasizing unique outputs per input, not requiring unique inputs per output or straight lines. A common error is thinking all outputs must be different (wrong—different x can share y) or that functions must be one-to-one (injective), which is a special type but not required. To check relations like this, (1) identify inputs (x-values or domain), (2) check each input (does it map to one output or multiple?), applicable across representations. Common mistakes include confusing 'same y for different x' as a violation (that's okay—many-to-one allowed) or claiming equations always functions (implicit relations like x²+y²=1 aren't—solving for y gives ±√(1-x²), two outputs).

This question tests understanding of the function definition: each input has exactly one output (one y per x), not allowing one input with multiple outputs. A function assigns exactly one output to each input: {(1,3),(2,5),(3,7)} is a function (x=1→3, x=2→5, x=3→7, each input once with one output), but {(2,3),(2,5),(4,7)} is NOT (x=2 maps to both 3 and 5, violates rule—input 2 has two outputs); graphically, the vertical line test applies (if any vertical line hits the graph more than once, it's not a function—that x has multiple y values); equations like y=2x+1 are functions (each x input gives exactly one y output by computation). In this set of ordered pairs {(2,3),(2,5),(4,7)}, the inputs are 2 and 4, but input 2 appears twice mapping to different outputs 3 and 5, which is a violation. This relation is not a function based on the one-output rule because input 2 does not have exactly one output, as it maps to two different values. A common error is thinking that different y-values for the same x might be okay or missing that x=2 appears twice with different y-values, or incorrectly believing no repeated y-values are required for functions (but that's not the issue here). To check relations like this, (1) identify inputs (x-values or domain), (2) check each input (does it map to one output or multiple?), and (5) for ordered pairs, look for the same x with different y (violation). Common mistakes include confusing 'same y for different x' as a violation (that's okay—many-to-one allowed) or missing repeated x in the pairs.

This question tests understanding of the function definition: each input has exactly one output (one y per x), not allowing one input with multiple outputs. A function assigns exactly one output to each input: {(1,3),(2,5),(3,7)} is a function (x=1→3, x=2→5, x=3→7, each input once with one output), but {(2,3),(2,5),(4,7)} is NOT (x=2 maps to both 3 and 5, violates rule—input 2 has two outputs); graphically, the vertical line test applies (if any vertical line hits the graph more than once, it's not a function—that x has multiple y values); equations like y=2x+1 are functions (each x input gives exactly one y output by computation). In this set of ordered pairs {(2,3),(2,5),(4,7)}, the inputs are 2 and 4, but input 2 appears twice mapping to different outputs 3 and 5, which is a violation. This relation is not a function based on the one-output rule because input 2 does not have exactly one output, as it maps to two different values. A common error is thinking that different y-values for the same x might be okay or missing that x=2 appears twice with different y-values, or incorrectly believing no repeated y-values are required for functions (but that's not the issue here). To check relations like this, (1) identify inputs (x-values or domain), (2) check each input (does it map to one output or multiple?), and (5) for ordered pairs, look for the same x with different y (violation). Common mistakes include confusing 'same y for different x' as a violation (that's okay—many-to-one allowed) or missing repeated x in the pairs.

This question tests understanding of the function definition: each input has exactly one output (one y per x), not allowing one input with multiple outputs. A function assigns exactly one output to each input: {(1,3),(2,5),(3,7)} is a function (x=1→3, x=2→5, x=3→7, each input once with one output), but {(2,3),(2,5),(4,7)} is NOT (x=2 maps to both 3 and 5, violates rule—input 2 has two outputs); graphically, the vertical line test applies (if any vertical line hits the graph more than once, it's not a function—that x has multiple y values); equations like y=2x+1 are functions (each x input gives exactly one y output by computation). In this graph with points (0,2), (0,-2), (1,1), and (-1,1), input x=0 maps to both 2 and -2, which is a violation, and the vertical line at x=0 intersects twice. This relation is not a function based on the one-output rule because input 0 does not have exactly one output. A common error is applying the vertical line test horizontally (wrong axis) or thinking no repeated y-values are needed (but here the issue is multiple y for same x). To check relations like this, (1) identify inputs (x-values or domain), (2) check each input (does it map to one output or multiple?), and (4) for graphs, use the vertical line test (imagine a vertical line sliding across; it hits twice at x=0). Common mistakes include incorrectly applying the vertical test as horizontal (wrong axis) or confusing 'same y for different x' as a violation (that's okay—many-to-one allowed, but not the case here).

This question tests understanding of the function definition: each input has exactly one output (one y per x), not allowing one input with multiple outputs. A function assigns exactly one output to each input: {(1,3),(2,5),(3,7)} is a function (x=1→3, x=2→5, x=3→7, each input once with one output), but {(2,3),(2,5),(4,7)} is NOT (x=2 maps to both 3 and 5, violates rule—input 2 has two outputs); graphically, the vertical line test applies (if any vertical line hits the graph more than once, it's not a function—that x has multiple y values); equations like y=2x+1 are functions (each x input gives exactly one y output by computation). This question asks for the core property, which is that each input has exactly one output, distinguishing functions from general relations where inputs might have multiple or no outputs. The correct statement aligns with the one-output rule, emphasizing unique outputs per input, not requiring unique inputs per output or straight lines. A common error is thinking all outputs must be different (wrong—different x can share y) or that functions must be one-to-one (injective), which is a special type but not required. To check relations like this, (1) identify inputs (x-values or domain), (2) check each input (does it map to one output or multiple?), applicable across representations. Common mistakes include confusing 'same y for different x' as a violation (that's okay—many-to-one allowed) or claiming equations always functions (implicit relations like x²+y²=1 aren't—solving for y gives ±√(1-x²), two outputs).

This question tests understanding of the function definition: each input has exactly one output (one y per x), not allowing one input with multiple outputs. A function assigns exactly one output to each input: {(1,3),(2,5),(3,7)} is a function (x=1→3, x=2→5, x=3→7, each input once with one output), but {(2,3),(2,5),(4,7)} is NOT (x=2 maps to both 3 and 5, violates rule—input 2 has two outputs); graphically, the vertical line test applies (if any vertical line hits the graph more than once, it's not a function—that x has multiple y values); equations like y=2x+1 are functions (each x input gives exactly one y output by computation). In this set of ordered pairs {(1,6),(2,12),(3,18),(4,24)}, the inputs are 1, 2, 3, and 4, each appearing once and mapping to a single output: 1→6, 2→12, 3→18, 4→24, with no input having multiple outputs. This relation is a function based on the one-output rule because every input has exactly one corresponding output, and there are no duplicates in the x-values with differing y-values. A common error is thinking repeated y-values violate the function rule (wrong—different x can share y, but here y-values are all different anyway), or mistakenly believing functions require the same output for all inputs. To check relations like this, (1) identify inputs (x-values or domain), (2) check each input (does it map to one output or multiple?), and (5) for ordered pairs, look for the same x with different y (which would be a violation). Common mistakes include confusing 'same y for different x' as a violation (that's okay—many-to-one allowed) or claiming equations always functions (implicit relations like x²+y²=1 aren't—solving for y gives ±√(1-x²), two outputs).

This question tests understanding of the function definition: each input has exactly one output (one y per x), not allowing one input with multiple outputs, and here applies it by evaluating a given function rule. A function assigns exactly one output to each input: {(1,3),(2,5),(3,7)} is a function (x=1→3, x=2→5, x=3→7, each input once with one output), but {(2,3),(2,5),(4,7)} is NOT (x=2 maps to both 3 and 5, violates rule—input 2 has two outputs); graphically, the vertical line test applies (if any vertical line hits the graph more than once, it's not a function—that x has multiple y values); equations like y=2x+1 are functions (each x input gives exactly one y output by computation). For the function f(x)=2x+1, plugging in x=3 gives f(3)=23+1=6+1=7, so the output is 7, demonstrating the unique output for that input. This confirms the function concept as it follows the one-output rule, producing exactly one value for each input via the equation. A common error is miscalculating the expression, like doing 23=6 without adding 1, or thinking functions can't have linear rules. To check or evaluate functions like this, (1) identify the input (here x=3), (2) apply the rule to find the single output, ensuring it maps to one value. Common mistakes include claiming equations always functions (implicit relations like x²+y²=1 aren't—solving for y gives ±√(1-x²), two outputs) or arithmetic errors in computation.

This question tests understanding of the function definition: each input has exactly one output (one y per x), not allowing one input with multiple outputs. A function assigns exactly one output to each input: {(1,3),(2,5),(3,7)} is a function (x=1→3, x=2→5, x=3→7, each input once with one output), but {(2,3),(2,5),(4,7)} is NOT (x=2 maps to both 3 and 5, violates rule—input 2 has two outputs); graphically, the vertical line test applies (if any vertical line hits the graph more than once, it's not a function—that x has multiple y values), and equations like y=2x+1 are functions (each x input gives exactly one y output by computation). For f(x)=2x+1, input x=3 gives output 2*3 + 1 = 7, which is exactly one output with no violations. Therefore, this is a function based on the one-output rule, and f(3)=7. A common error is miscalculating the arithmetic, like forgetting to add 1 or doubling incorrectly. To check equations like this, (1) identify inputs (x values or domain), (2) compute the output for each (does it give one value?), confirming it's a function. Mistakes include claiming equations always functions (implicit relations like x²+y²=1 aren't—solving for y gives ±√(1-x²), two outputs), or confusing function evaluation with other concepts.