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Four relations are shown.

Relation A has two columns x and y. When x is 4, y is 4. When x is 5, y is 5. When x is 5, y is 0. When x is 7, y is 8. When x is 10, y is 1. Relation B has a coordinate plane with a U-shaped curve open to the right with its vertex at (-4.2, 0). Relation C shows the equation 4x plus 53 equals to y. Relation D shows the equation y equal to 8x exponential 2.

Drag the name of each relation to identify whether the relation is a function or not a function.
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function --------------------- not a function

relation a, relation b, relation c, relation d

1 answer

To determine whether each relation is a function, we can use the vertical line test and analyze the definitions of functions. A relation is a function if, for every input (x-value), there is exactly one output (y-value).

  1. Relation A: This relation has the following pairs:

    • (4, 4)
    • (5, 5)
    • (5, 0)
    • (7, 8)
    • (10, 1)

    Here, the x-value 5 corresponds to two different y-values (5 and 0). Therefore, Relation A is not a function.

  2. Relation B: This relation is described as a U-shaped curve (a parabola) that opens to the right with its vertex at (-4.2, 0). For each x-value in this graph, there will be exactly one y-value (as it is a function of x). Hence, Relation B is a function.

  3. Relation C: The equation given is \( y = 4x + 53 \), which is a linear equation. For every x-value, there is exactly one corresponding y-value. Therefore, Relation C is a function.

  4. Relation D: The equation is \( y = 8x^2 \), which is also a quadratic function (a parabola opening upwards). For every x-value, there will only be one corresponding y-value. Therefore, Relation D is a function.

In summary:

  • Function: Relation B, Relation C, Relation D
  • Not a function: Relation A