To determine whether the relation between Celsius temperature [tex]\(c\)[/tex] and Fahrenheit temperature [tex]\(f\)[/tex] is a function, we need to analyze the given statements. Let's go through each one step by step.
1. It is a function because -40°C is paired with -40°F.
- This statement suggests that there exists at least one pair [tex]\((-40, -40)\)[/tex] where [tex]\(-40^\circ C\)[/tex] corresponds to [tex]\(-40^\circ F\)[/tex]. While this indicates a specific instance of a temperature pair, it alone is not sufficient to determine whether the entire relation is a function. A function requires that every input has exactly one output.
2. It is a function because every Celsius temperature is associated with only one Fahrenheit temperature.
- This is a key characteristic of a function. We need to check if the formula [tex]\(f = \frac{9}{5}c + 32\)[/tex] uniquely determines [tex]\(f\)[/tex] for every [tex]\(c\)[/tex]. Given the equation and solving for different values of [tex]\(c\)[/tex], each Celsius input [tex]\(c\)[/tex] leads to a unique Fahrenheit output [tex]\(f\)[/tex]. For example,
- if [tex]\(c = -40^\circ\)[/tex], then [tex]\(f = \frac{9}{5}(-40) + 32 = -40^\circ\)[/tex],
- if [tex]\(c = 0^\circ\)[/tex], then [tex]\(f = \frac{9}{5}(0) + 32 = 32^\circ\)[/tex],
- if [tex]\(c = 100^\circ\)[/tex], then [tex]\(f = \frac{9}{5}(100) + 32 = 212^\circ\)[/tex].
Since each input has a unique output, this statement is correct.
3. It is not a function because 0°C is not paired with 0°F.
- This statement is misleading. The fact that [tex]\(0^\circ C\)[/tex] is paired with [tex]\(32^\circ F\)[/tex] (as calculated above) does not imply the relation is not a function. A function merely requires that each input temperature has a unique corresponding output, regardless of what those outputs specifically are. Thus, this statement is incorrect.
4. It is not a function because some Celsius temperatures cannot be associated with a Fahrenheit temperature.
- This statement suggests that some Celsius inputs might not produce a corresponding Fahrenheit output. However, the conversion formula [tex]\(f = \frac{9}{5}c + 32\)[/tex] is defined for all real numbers. Therefore, any Celsius temperature [tex]\(c\)[/tex] can be converted to a Fahrenheit temperature [tex]\(f\)[/tex] using this formula. This statement is also incorrect.
The best statement that describes if the relation ([tex]\(c, f\)[/tex]) is a function is:
"It is a function because every Celsius temperature is associated with only one Fahrenheit temperature."
