Answer :
To determine the range of the function given in the problem, let's break down the details step by step:
1. Identify the Protein Content from Different Sources:
- Livia eats a chicken drumstick which provides 11 grams of protein.
- Livia also eats [tex]\(x\)[/tex] cheese sticks, each cheese stick providing 7 grams of protein.
2. Formulate the Equation:
- The total protein [tex]\(y\)[/tex] that Livia consumes can be expressed as:
[tex]\[ y = 11 + 7x \][/tex]
where:
- [tex]\(y\)[/tex] is the total grams of protein.
- [tex]\(x\)[/tex] is the number of cheese sticks consumed.
3. Explore the Possible Values for [tex]\(x\)[/tex]:
- Since Livia can eat any number of cheese sticks, including fractions, [tex]\(x\)[/tex] is any non-negative real number (i.e., [tex]\(x \geq 0\)[/tex]).
4. Determine the Minimum Value of [tex]\(y\)[/tex]:
- If Livia eats no cheese sticks ([tex]\(x = 0\)[/tex]):
[tex]\[ y = 11 + 7(0) = 11 \][/tex]
- Therefore, the minimum value of [tex]\(y\)[/tex] is 11 grams of protein.
5. Analyze the Range of [tex]\(y\)[/tex]:
- As [tex]\(x\)[/tex] increases (considering it can be any non-negative real number), [tex]\(y\)[/tex] will continue to increase because [tex]\(7x\)[/tex] adds to the initial 11 grams.
- There is no upper bound to [tex]\(y\)[/tex] since [tex]\(x\)[/tex] can become arbitrarily large.
6. Conclusion on the Range of the Function:
- Since [tex]\(x\)[/tex] can be any non-negative real number, and the minimum total protein intake [tex]\(y\)[/tex] starts at 11 grams and can go up indefinitely, we conclude that:
[tex]\[ y \text{ can be any real number greater than or equal to 11} \][/tex]
Therefore, the correct answer is:
[tex]\[ \text{The range of the function is all real numbers greater than or equal to 11} \][/tex]
1. Identify the Protein Content from Different Sources:
- Livia eats a chicken drumstick which provides 11 grams of protein.
- Livia also eats [tex]\(x\)[/tex] cheese sticks, each cheese stick providing 7 grams of protein.
2. Formulate the Equation:
- The total protein [tex]\(y\)[/tex] that Livia consumes can be expressed as:
[tex]\[ y = 11 + 7x \][/tex]
where:
- [tex]\(y\)[/tex] is the total grams of protein.
- [tex]\(x\)[/tex] is the number of cheese sticks consumed.
3. Explore the Possible Values for [tex]\(x\)[/tex]:
- Since Livia can eat any number of cheese sticks, including fractions, [tex]\(x\)[/tex] is any non-negative real number (i.e., [tex]\(x \geq 0\)[/tex]).
4. Determine the Minimum Value of [tex]\(y\)[/tex]:
- If Livia eats no cheese sticks ([tex]\(x = 0\)[/tex]):
[tex]\[ y = 11 + 7(0) = 11 \][/tex]
- Therefore, the minimum value of [tex]\(y\)[/tex] is 11 grams of protein.
5. Analyze the Range of [tex]\(y\)[/tex]:
- As [tex]\(x\)[/tex] increases (considering it can be any non-negative real number), [tex]\(y\)[/tex] will continue to increase because [tex]\(7x\)[/tex] adds to the initial 11 grams.
- There is no upper bound to [tex]\(y\)[/tex] since [tex]\(x\)[/tex] can become arbitrarily large.
6. Conclusion on the Range of the Function:
- Since [tex]\(x\)[/tex] can be any non-negative real number, and the minimum total protein intake [tex]\(y\)[/tex] starts at 11 grams and can go up indefinitely, we conclude that:
[tex]\[ y \text{ can be any real number greater than or equal to 11} \][/tex]
Therefore, the correct answer is:
[tex]\[ \text{The range of the function is all real numbers greater than or equal to 11} \][/tex]
