To determine whether the relation ( [tex]\( c, f \)[/tex] ) defined by the equation [tex]\( f = \frac{9}{5} c + 32 \)[/tex] is a function, we must assess if each input (Celsius temperature, [tex]\( c \)[/tex]) has exactly one output (Fahrenheit temperature, [tex]\( f \)[/tex]).
Let's analyze each of the given statements:
1. "It is a function because [tex]\(-40^{\circ} C\)[/tex] is paired with [tex]\(-40^{\circ} F\)[/tex]."
- This statement provides a specific example but does not address the overall nature of the relation for all [tex]\( c \)[/tex].
2. "It is a function because every Celsius temperature is associated with only one Fahrenheit temperature."
- This statement directly addresses the definition of a function: each input (Celsius temperature, [tex]\( c \)[/tex]) should map to exactly one output (Fahrenheit temperature, [tex]\( f \)[/tex]).
- According to the equation [tex]\( f = \frac{9}{5} c + 32 \)[/tex], we're given a linear relationship. For any given [tex]\( c \)[/tex], there will be a unique [tex]\( f \)[/tex] calculated by this formula.
3. "It is not a function because [tex]\(0^{\circ} C\)[/tex] is not paired with [tex]\(0^{\circ} F\)[/tex]."
- This statement is incorrect because it misinterprets the concept of a function. The fact that [tex]\(0^{\circ} C\)[/tex] does not yield [tex]\(0^{\circ} F\)[/tex] (in fact, it yields [tex]\(32^{\circ} F\)[/tex]) does not mean the relation isn't a function.
4. "It is not a function because some Celsius temperatures cannot be associated with a Fahrenheit temperature."
- This statement is incorrect because every Celsius temperature [tex]\( c \)[/tex] can indeed be converted into a Fahrenheit temperature [tex]\( f \)[/tex] using the given linear equation.
Given this analysis, the correct statement that best describes the relationship ( [tex]\( c, f \)[/tex] ) is:
"It is a function because every Celsius temperature is associated with only one Fahrenheit temperature."
Therefore, the correct option is:
Option 2.
