Answer :
To determine which function has a domain of \((-\infty, \infty)\) and a range of \((-\infty, 4]\), let's analyze each of the given functions:
1. \( f(x) = 2^x + 4 \):
- Domain: The function \(2^x\) is defined for all real numbers \(x\).
- Range: The term \(2^x\) will yield positive values for any real number \(x\), starting from \(2^0=1\). So, \(2^x \geq 0\). When we add 4, the smallest value this function can take is \(2^0 + 4 = 5\). Since \(2^x\) can grow without bound as \(x \to \infty\), the range of \(f(x) = 2^x + 4\) is \((4, \infty)\).
- Hence, the range does not match \((-\infty, 4]\).
2. \( f(x) = -x^2 + 4 \):
- Domain: The function \( -x^2 \) is defined for all real numbers \(x\).
- Range: Since \( -x^2 \) will always be non-positive (i.e., \( -x^2 \leq 0 \)), and \( -x^2 \) attains its maximum value of 0 when \( x = 0 \). So, the maximum value of \( -x^2 + 4 \) is 4 and the function can take any value less than or equal to 4. This implies the range of \(f(x) = -x^2 + 4\) is \((-\infty, 4]\).
- This matches the specified range.
3. \( f(x) = x + 4 \):
- Domain: The function \(x + 4 \) is a linear polynomial defined for all real numbers \(x\).
- Range: Since \( x \) can take any real number, the output \( x + 4 \) can also take any real number.
- Hence, the range is \((-\infty, \infty)\), which does not match \((-\infty, 4]\).
4. \( f(x) = -4x \):
- Domain: The function \(-4x \) is a linear polynomial defined for all real numbers \(x\).
- Range: Since \( x \) can take any real number, the output \( -4x \) can also take any real number.
- Hence, the range is \((-\infty, \infty)\), which does not match \((-\infty, 4]\).
After analyzing the functions, we see that the function \(f(x) = -x^2 + 4\) has the desired domain and range. Therefore, the correct answer is:
[tex]\[ \boxed{f(x) = -x^2 + 4} \][/tex]
1. \( f(x) = 2^x + 4 \):
- Domain: The function \(2^x\) is defined for all real numbers \(x\).
- Range: The term \(2^x\) will yield positive values for any real number \(x\), starting from \(2^0=1\). So, \(2^x \geq 0\). When we add 4, the smallest value this function can take is \(2^0 + 4 = 5\). Since \(2^x\) can grow without bound as \(x \to \infty\), the range of \(f(x) = 2^x + 4\) is \((4, \infty)\).
- Hence, the range does not match \((-\infty, 4]\).
2. \( f(x) = -x^2 + 4 \):
- Domain: The function \( -x^2 \) is defined for all real numbers \(x\).
- Range: Since \( -x^2 \) will always be non-positive (i.e., \( -x^2 \leq 0 \)), and \( -x^2 \) attains its maximum value of 0 when \( x = 0 \). So, the maximum value of \( -x^2 + 4 \) is 4 and the function can take any value less than or equal to 4. This implies the range of \(f(x) = -x^2 + 4\) is \((-\infty, 4]\).
- This matches the specified range.
3. \( f(x) = x + 4 \):
- Domain: The function \(x + 4 \) is a linear polynomial defined for all real numbers \(x\).
- Range: Since \( x \) can take any real number, the output \( x + 4 \) can also take any real number.
- Hence, the range is \((-\infty, \infty)\), which does not match \((-\infty, 4]\).
4. \( f(x) = -4x \):
- Domain: The function \(-4x \) is a linear polynomial defined for all real numbers \(x\).
- Range: Since \( x \) can take any real number, the output \( -4x \) can also take any real number.
- Hence, the range is \((-\infty, \infty)\), which does not match \((-\infty, 4]\).
After analyzing the functions, we see that the function \(f(x) = -x^2 + 4\) has the desired domain and range. Therefore, the correct answer is:
[tex]\[ \boxed{f(x) = -x^2 + 4} \][/tex]
