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Given the domain [tex]$\{-4, 0, 5\}$[/tex], what is the range for the relation [tex]$12x + 6y = 24$[/tex]?

A. [tex]$\{-4, 4, 14\}$[/tex]
B. [tex]$\{-12, -4, 6\}$[/tex]
C. [tex]$\{2, 4, 9\}$[/tex]
D. [tex]$\{12, 4, -6\}$[/tex]

Sagot :

GinnyAnswer
To find the range for the given relation \( 12x + 6y = 24 \) when the domain is \( \{-4, 0, 5\} \), we need to solve for \( y \) in terms of \( x \).

Given the equation:
[tex]\[ 12x + 6y = 24 \][/tex]

First, isolate \( y \):

[tex]\[ 6y = 24 - 12x \][/tex]
[tex]\[ y = \frac{24 - 12x}{6} \][/tex]
[tex]\[ y = 4 - 2x \][/tex]

Now, we will substitute each value of the domain into this equation to find the corresponding \( y \)-values, which will give us the range.

1. When \( x = -4 \):
[tex]\[ y = 4 - 2(-4) \][/tex]
[tex]\[ y = 4 + 8 \][/tex]
[tex]\[ y = 12 \][/tex]

2. When \( x = 0 \):
[tex]\[ y = 4 - 2(0) \][/tex]
[tex]\[ y = 4 \][/tex]

3. When \( x = 5 \):
[tex]\[ y = 4 - 2(5) \][/tex]
[tex]\[ y = 4 - 10 \][/tex]
[tex]\[ y = -6 \][/tex]

So, the \( y \)-values (the range) corresponding to the domain \(\{-4, 0, 5\}\) are:

[tex]\[ \{12, 4, -6\} \][/tex]

Thus, the correct answer is:
D. [tex]\( \{12, 4, -6\} \)[/tex]
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