Question
Function g is defined by g(x) = x ^ z + 4x + 3 What are the solutions to g(x) = 0
x = - 4 and x = 0
x = - 12 and x = - 9
25 x = 1
x = - 3 and x = - 1
x = - 4 and x = 0
x = - 12 and x = - 9
25 x = 1
x = - 3 and x = - 1
Answers
Answered by
GPT-4o mini
To find the solutions to the equation \( g(x) = 0 \) where \( g(x) = x^z + 4x + 3 \), we need more information on the variable \( z \). However, we can analyze the given options with the assumption that \( z \) could take certain values.
Let’s verify the options step by step to see if any of them satisfy \( g(x) = 0 \).
1. **Option x = -4 and x = 0**:
- For \( x = -4 \):
\( g(-4) = (-4)^z + 4(-4) + 3 = (-4)^z - 16 + 3 = (-4)^z - 13 \)
This would be zero if \( (-4)^z = 13 \), depending on \( z \).
- For \( x = 0 \):
\( g(0) = 0^z + 4(0) + 3 = 3 \)
This does not equal zero.
2. **Option x = -12 and x = -9**:
- For \( x = -12 \):
\( g(-12) = (-12)^z + 4(-12) + 3 = (-12)^z - 48 + 3 = (-12)^z - 45 \)
- For \( x = -9 \):
\( g(-9) = (-9)^z + 4(-9) + 3 = (-9)^z - 36 + 3 = (-9)^z - 33 \)
- No particular solutions can be claimed without knowing \( z \).
3. **Option x = 1**:
- For \( x = 1 \):
\( g(1) = 1^z + 4(1) + 3 = 1 + 4 + 3 = 8 \)
This does not equal zero.
4. **Option x = -3 and x = -1**:
- For \( x = -3 \):
\( g(-3) = (-3)^z + 4(-3) + 3 = (-3)^z - 12 + 3 = (-3)^z - 9 \)
- For \( x = -1 \):
\( g(-1) = (-1)^z + 4(-1) + 3 = (-1)^z - 4 + 3 = (-1)^z - 1 \)
- Again, no zeros can be confirmed without knowing \( z \).
### Conclusion
To determine the actual roots of \( g(x) \) you need to know the specific value of \( z \) in order to correctly evaluate \( g(x) \) for specific points in the options provided. None of the options can be conclusively identified as solutions to \( g(x) = 0 \) without more information. If \( z \) is provided, I can assist with finding the specific roots more effectively.
Let’s verify the options step by step to see if any of them satisfy \( g(x) = 0 \).
1. **Option x = -4 and x = 0**:
- For \( x = -4 \):
\( g(-4) = (-4)^z + 4(-4) + 3 = (-4)^z - 16 + 3 = (-4)^z - 13 \)
This would be zero if \( (-4)^z = 13 \), depending on \( z \).
- For \( x = 0 \):
\( g(0) = 0^z + 4(0) + 3 = 3 \)
This does not equal zero.
2. **Option x = -12 and x = -9**:
- For \( x = -12 \):
\( g(-12) = (-12)^z + 4(-12) + 3 = (-12)^z - 48 + 3 = (-12)^z - 45 \)
- For \( x = -9 \):
\( g(-9) = (-9)^z + 4(-9) + 3 = (-9)^z - 36 + 3 = (-9)^z - 33 \)
- No particular solutions can be claimed without knowing \( z \).
3. **Option x = 1**:
- For \( x = 1 \):
\( g(1) = 1^z + 4(1) + 3 = 1 + 4 + 3 = 8 \)
This does not equal zero.
4. **Option x = -3 and x = -1**:
- For \( x = -3 \):
\( g(-3) = (-3)^z + 4(-3) + 3 = (-3)^z - 12 + 3 = (-3)^z - 9 \)
- For \( x = -1 \):
\( g(-1) = (-1)^z + 4(-1) + 3 = (-1)^z - 4 + 3 = (-1)^z - 1 \)
- Again, no zeros can be confirmed without knowing \( z \).
### Conclusion
To determine the actual roots of \( g(x) \) you need to know the specific value of \( z \) in order to correctly evaluate \( g(x) \) for specific points in the options provided. None of the options can be conclusively identified as solutions to \( g(x) = 0 \) without more information. If \( z \) is provided, I can assist with finding the specific roots more effectively.