To determine the range of the function, we need to analyze the total number of grams of protein [tex]\( y \)[/tex] that Livia consumes based on the number of cheese sticks [tex]\( x \)[/tex] she eats.
Let's break down the given information step by step:
1. Chicken Drumstick Protein: The protein from the chicken drumstick is always 11 grams, which is a constant value.
2. Cheese Stick Protein: Each cheese stick contains 7 grams of protein. Therefore, if Livia eats [tex]\( x \)[/tex] cheese sticks, the protein from cheese sticks would be [tex]\( 7x \)[/tex] grams.
3. Total Protein: The total protein [tex]\( y \)[/tex] Livia consumes is the sum of the protein from the chicken drumstick and the cheese sticks. Mathematically, this can be expressed as:
[tex]\[
y = 11 + 7x
\][/tex]
Now, let's use this relationship to determine the quantities in the table:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
y = 11 + 7(0) = 11 \text{ grams}
\][/tex]
- For [tex]\( x = 2.5 \)[/tex]:
[tex]\[
y = 11 + 7(2.5) = 11 + 17.5 = 28.5 \text{ grams}
\][/tex]
- For [tex]\( x = 5 \)[/tex]:
[tex]\[
y = 11 + 7(5) = 11 + 35 = 46 \text{ grams}
\][/tex]
- For [tex]\( x = 7 \)[/tex]:
[tex]\[
y = 11 + 7(7) = 11 + 49 = 60 \text{ grams}
\][/tex]
These values match the values given in the table:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
0 & 11 \\
\hline
2.5 & 28.5 \\
\hline
5 & 46 \\
\hline
7 & 60 \\
\hline
\end{tabular}
\][/tex]
Considering the formula [tex]\( y = 11 + 7x \)[/tex], and knowing that [tex]\( x \)[/tex] can be any non-negative real number (since Livia can eat parts of cheese sticks), we see that:
- When [tex]\( x = 0 \)[/tex], the minimum value of [tex]\( y \)[/tex] is 11 grams.
- As [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] will continue to increase linearly.
Thus, the smallest value [tex]\( y \)[/tex] can take is 11 grams when [tex]\( x = 0 \)[/tex]. There is no upper limit for [tex]\( y \)[/tex] because [tex]\( x \)[/tex] can grow indefinitely large.
Therefore, the range of [tex]\( y \)[/tex] is:
[tex]\[
\text{All real numbers greater than or equal to 11}
\][/tex]
This matches the option: all real numbers greater than or equal to 11.
