Let's analyze each function one by one to determine their domains and ranges.
1. Function: [tex]\( f(x) = -4x \)[/tex]
- Domain: The domain of a linear function is all real numbers. Therefore, [tex]\( \text{Domain} = (-\infty, \infty) \)[/tex].
- Range: The range of a linear function that goes from negative to positive infinity is also all real numbers. Therefore, [tex]\( \text{Range} = (-\infty, \infty) \)[/tex].
2. Function: [tex]\( f(x) = x + 4 \)[/tex]
- Domain: The domain of a linear function is all real numbers. Therefore, [tex]\( \text{Domain} = (-\infty, \infty) \)[/tex].
- Range: The range of a linear function that goes from negative to positive infinity is also all real numbers. Therefore, [tex]\( \text{Range} = (-\infty, \infty) \)[/tex].
3. Function: [tex]\( f(x) = 2^x + 4 \)[/tex]
- Domain: The domain of an exponential function is all real numbers. Therefore, [tex]\( \text{Domain} = (-\infty, \infty) \)[/tex].
- Range: The output values of [tex]\(2^x\)[/tex] are always positive and start from 0 (but never actually reach zero). Thus, [tex]\(f(x) = 2^x + 4\)[/tex] shifts the entire range by 4 units up. Therefore, [tex]\( \text{Range} = (4, \infty) \)[/tex].
4. Function: [tex]\( f(x) = -x^2 + 4 \)[/tex]
- Domain: The domain of a polynomial function is all real numbers. Therefore, [tex]\( \text{Domain} = (-\infty, \infty) \)[/tex].
- Range: Since [tex]\(-x^2\)[/tex] is always non-positive and reaches its maximum value at 0 (when [tex]\(x = 0\)[/tex]), adding 4 shifts the vertex to 4. Therefore, the highest value is 4, and it can go down to negative infinity. Thus, [tex]\( \text{Range} = (-\infty, 4] \)[/tex].
Given the analysis above, the only function that has the domain [tex]\((-\infty, \infty)\)[/tex] and range [tex]\((-\infty, 4]\)[/tex] is:
[tex]\[ f(x) = -x^2 + 4 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{f(x) = -x^2 + 4} \][/tex]
Thus, the correct function index from the provided list is:
[tex]\[ \boxed{4} \][/tex]
