Let's analyze the situation step by step:
1. Given Data:
- Each chicken drumstick has 11 grams of protein.
- Each cheese stick has 7 grams of protein.
- Livia can eat [tex]\( x \)[/tex] cheese sticks, where [tex]\( x \)[/tex] can be any real number (not necessarily an integer).
2. Expression for Total Protein:
- The total amount of protein [tex]\( y \)[/tex] Livia will consume if she eats [tex]\( x \)[/tex] cheese sticks is given by the expression:
[tex]\[
y = 11 + 7x
\][/tex]
3. Understanding the Range:
- The range is the set of all possible values that [tex]\( y \)[/tex] can take.
- When [tex]\( x = 0 \)[/tex], Livia eats no cheese sticks, so the total protein is only from the chicken drumstick:
[tex]\[
y = 11 + 7 \cdot 0 = 11
\][/tex]
- As [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] increases proportionally because [tex]\( y = 11 + 7x \)[/tex] is a linear function with a positive slope (7).
4. Analyzing Values:
- If [tex]\( x \)[/tex] is any positive real number, [tex]\( y \)[/tex] will be greater than 11.
- Since [tex]\( x \)[/tex] can be any non-negative real number (including decimals and fractions), the smallest value [tex]\( y \)[/tex] can take is 11 (when [tex]\( x = 0 \)[/tex]).
- There is no upper limit to the value of [tex]\( y \)[/tex] because there is no upper limit to [tex]\( x \)[/tex]; as [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] also increases without bound.
5. Conclusion:
- Given that [tex]\( x \)[/tex] can be any real number greater than or equal to 0, [tex]\( y \)[/tex] can be any real number greater than or equal to 11.
Therefore, the range of the function is:
[tex]\[ \text{all real numbers greater than or equal to 11} \][/tex]
Which corresponds to the third option provided:
[tex]\[ \textbf{all real numbers greater than or equal to 11} \][/tex]
