Answer :
Let's analyze each feature one by one to evaluate the correctness of the student's findings for the function \( f(x) = 2^x - 4 \).
### 1. Increasing:
The function \( f(x) = 2^x - 4 \) is indeed increasing. This is because the exponential function \( 2^x \) is always increasing, and subtracting a constant (in this case, 4) does not change the increasing nature of the function.
Verdict: Correct
### 2. Positive for \( x > -3 \):
To determine where the function \( f(x) = 2^x - 4 \) is positive, we solve:
[tex]\[ 2^x - 4 > 0 \][/tex]
[tex]\[ 2^x > 4 \][/tex]
Taking the logarithm (base 2) on both sides:
[tex]\[ x > 2 \][/tex]
Therefore, the function \( f(x) = 2^x - 4 \) is positive for \( x > 2 \), not for \( x > -3 \).
Verdict: Incorrect
### 3. Negative for \( x < 2 \):
To determine where the function \( f(x) = 2^x - 4 \) is negative, we solve:
[tex]\[ 2^x - 4 < 0 \][/tex]
[tex]\[ 2^x < 4 \][/tex]
Taking the logarithm (base 2) on both sides:
[tex]\[ x < 2 \][/tex]
Therefore, the function \( f(x) = 2^x - 4 \) is negative for \( x < 2 \).
Verdict: Correct
### 4. As \( x \) approaches negative infinity, \( f(x) \) approaches -4:
For very large negative values of \( x \), the value \( 2^x \) approaches 0. Thus:
[tex]\[ f(x) = 2^x - 4 \][/tex]
[tex]\[ \lim_{x \to -\infty} (2^x - 4) = -4 \][/tex]
Verdict: Correct
### 5. As \( x \) approaches positive infinity, \( f(x) \) approaches 4:
For very large positive values of \( x \), the value of \( 2^x \) grows without bound. Thus:
[tex]\[ f(x) = 2^x - 4 \][/tex]
[tex]\[ \lim_{x \to \infty} (2^x - 4) = \infty \][/tex]
Therefore, as \( x \) approaches positive infinity, \( f(x) \) approaches positive infinity, not 4.
Verdict: Incorrect
### Summary:
The student incorrectly identified two key features:
- The function is positive for \( x > 2 \), not \( x > -3 \).
- As \( x \) approaches positive infinity, \( f(x) \) approaches positive infinity, not 4.
The remaining key features identified by the student are correct.
### 1. Increasing:
The function \( f(x) = 2^x - 4 \) is indeed increasing. This is because the exponential function \( 2^x \) is always increasing, and subtracting a constant (in this case, 4) does not change the increasing nature of the function.
Verdict: Correct
### 2. Positive for \( x > -3 \):
To determine where the function \( f(x) = 2^x - 4 \) is positive, we solve:
[tex]\[ 2^x - 4 > 0 \][/tex]
[tex]\[ 2^x > 4 \][/tex]
Taking the logarithm (base 2) on both sides:
[tex]\[ x > 2 \][/tex]
Therefore, the function \( f(x) = 2^x - 4 \) is positive for \( x > 2 \), not for \( x > -3 \).
Verdict: Incorrect
### 3. Negative for \( x < 2 \):
To determine where the function \( f(x) = 2^x - 4 \) is negative, we solve:
[tex]\[ 2^x - 4 < 0 \][/tex]
[tex]\[ 2^x < 4 \][/tex]
Taking the logarithm (base 2) on both sides:
[tex]\[ x < 2 \][/tex]
Therefore, the function \( f(x) = 2^x - 4 \) is negative for \( x < 2 \).
Verdict: Correct
### 4. As \( x \) approaches negative infinity, \( f(x) \) approaches -4:
For very large negative values of \( x \), the value \( 2^x \) approaches 0. Thus:
[tex]\[ f(x) = 2^x - 4 \][/tex]
[tex]\[ \lim_{x \to -\infty} (2^x - 4) = -4 \][/tex]
Verdict: Correct
### 5. As \( x \) approaches positive infinity, \( f(x) \) approaches 4:
For very large positive values of \( x \), the value of \( 2^x \) grows without bound. Thus:
[tex]\[ f(x) = 2^x - 4 \][/tex]
[tex]\[ \lim_{x \to \infty} (2^x - 4) = \infty \][/tex]
Therefore, as \( x \) approaches positive infinity, \( f(x) \) approaches positive infinity, not 4.
Verdict: Incorrect
### Summary:
The student incorrectly identified two key features:
- The function is positive for \( x > 2 \), not \( x > -3 \).
- As \( x \) approaches positive infinity, \( f(x) \) approaches positive infinity, not 4.
The remaining key features identified by the student are correct.
