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Which of the following represent [tex]$y$[/tex] as a function of [tex]$x$[/tex]?

Select all that apply.

A. [tex]y = x^2[/tex]
B. [tex]y = x[/tex]
C. [tex]\{(0,5), (1,6), (1,9)\}[/tex]

Sagot :

GinnyAnsweridn
To determine whether each given relation represents [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex], we need to ensure that each input [tex]\( x \)[/tex] corresponds to exactly one output [tex]\( y \)[/tex]. Let's examine each option step by step:

1. Option [tex]\( y = x^2 \)[/tex]:

- This equation describes a parabola.
- For any given value of [tex]\( x \)[/tex], squaring it yields exactly one value of [tex]\( y \)[/tex].
- For example, if [tex]\( x = 2 \)[/tex], then [tex]\( y = 2^2 = 4 \)[/tex]. Similarly, if [tex]\( x = -2 \)[/tex], then [tex]\( y = (-2)^2 = 4 \)[/tex].
- Every input [tex]\( x \)[/tex] has a unique output [tex]\( y \)[/tex].

Therefore, [tex]\( y = x^2 \)[/tex] represents [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].

2. Option [tex]\( y = x \)[/tex]:

- This equation describes a straight line with a slope of 1.
- For any given value of [tex]\( x \)[/tex], the value of [tex]\( y \)[/tex] is exactly the same as [tex]\( x \)[/tex].
- For example, if [tex]\( x = 3 \)[/tex], then [tex]\( y = 3 \)[/tex]. Similarly, if [tex]\( x = -3 \)[/tex], then [tex]\( y = -3 \)[/tex].
- Every input [tex]\( x \)[/tex] has a unique output [tex]\( y \)[/tex].

Therefore, [tex]\( y = x \)[/tex] represents [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].

3. Option [tex]\( \{(0,5),(1,6),(1,9)\} \)[/tex]:

- This is a set of ordered pairs.
- For a relation to be a function, each [tex]\( x \)[/tex] value must be paired with exactly one [tex]\( y \)[/tex] value.
- In this relation, the input value [tex]\( x = 1 \)[/tex] is associated with two different [tex]\( y \)[/tex] values (6 and 9).
- This means that the input [tex]\( x = 1 \)[/tex] does not have a unique [tex]\( y \)[/tex] value.

Therefore, [tex]\( \{(0,5),(1,6),(1,9)\} \)[/tex] does not represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].

In conclusion, the relations [tex]\( y = x^2 \)[/tex] and [tex]\( y = x \)[/tex] represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex], while the set [tex]\( \{(0,5),(1,6),(1,9)\} \)[/tex] does not.
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