Question
The graph of function f is shown.
The graph of an exponential function passes through (minus 0.25, 10), (0, 6), (5, minus 2) also intercepts the x-axis at 1 unit.
Function g is represented by the table.
x -1 0 1 2 3
g(x) 15 3 0
Which statement correctly compares the two functions?
A.
They have different x- and y-intercepts but the same end behavior as x approaches ∞.
B.
They have the same x-intercept but different end behavior as x approaches ∞.
C.
They have the same x- and y-intercepts.
D.
They have the same y-intercept and the same end behavior as x approaches ∞.
The graph of an exponential function passes through (minus 0.25, 10), (0, 6), (5, minus 2) also intercepts the x-axis at 1 unit.
Function g is represented by the table.
x -1 0 1 2 3
g(x) 15 3 0
Which statement correctly compares the two functions?
A.
They have different x- and y-intercepts but the same end behavior as x approaches ∞.
B.
They have the same x-intercept but different end behavior as x approaches ∞.
C.
They have the same x- and y-intercepts.
D.
They have the same y-intercept and the same end behavior as x approaches ∞.
Answers
GPT-4o mini
To analyze the functions \( f \) and \( g \) based on the provided information:
1. **Exponential Function \( f \)**:
- It passes through the points \((-0.25, 10)\), \((0, 6)\), and \( (5, -2) \).
- The fact that it intercepts the x-axis at 1 indicates \( f(1) = 0 \).
- An exponential function typically has the end behavior where it approaches 0 as \( x \) approaches \(-\infty\) and increases without bound as \( x \) approaches \( \infty \).
- \( f \) has a y-intercept at \( f(0) = 6 \).
2. **Function \( g \)** (given as a table):
- The values are:
- \( g(-1) = 15 \)
- \( g(0) = 3 \)
- \( g(1) = 0 \)
- The function has an x-intercept at \( g(1) = 0 \).
- Since \( g(0) = 3 \), the y-intercept is \( 3 \).
- The values of \( g(x) \) are decreasing, but we don't have information about \( g(x) \) for \( x > 1 \) to determine its end behavior.
3. **Interceptions**:
- Both functions have the same x-intercept at \( x = 1 \).
- The y-intercepts are different: \( f \) has a y-intercept of \( 6 \) while \( g \) has a y-intercept of \( 3 \).
4. **End Behavior**:
- As \( x \) approaches \( ∞\):
- The function \( f \) (being an exponential function) will tend to \( ∞\).
- The behavior of \( g(x) \) as \( x \to ∞ \) cannot be determined from the table alone. We only have values for \( x = -1, 0, 1, 2, 3 \).
Based on this analysis, we can conclude:
- **Statement A** is incorrect because they have different y-intercepts.
- **Statement B** is correct in that they have the same x-intercept but we cannot conclude that their end behavior is different due to lack of information about \( g(x) \) as \( x \to ∞\).
- **Statement C** is incorrect as discussed above.
- **Statement D** is incorrect because they do not have the same y-intercept.
Thus, none of the statements perfectly describe the comparison, but the closest is **B**, assuming we assert that \( g \) behaves differently in end behavior (since we do not have sufficient information on that front). So, the answer is:
**B. They have the same x-intercept but different end behavior as x approaches ∞.**
1. **Exponential Function \( f \)**:
- It passes through the points \((-0.25, 10)\), \((0, 6)\), and \( (5, -2) \).
- The fact that it intercepts the x-axis at 1 indicates \( f(1) = 0 \).
- An exponential function typically has the end behavior where it approaches 0 as \( x \) approaches \(-\infty\) and increases without bound as \( x \) approaches \( \infty \).
- \( f \) has a y-intercept at \( f(0) = 6 \).
2. **Function \( g \)** (given as a table):
- The values are:
- \( g(-1) = 15 \)
- \( g(0) = 3 \)
- \( g(1) = 0 \)
- The function has an x-intercept at \( g(1) = 0 \).
- Since \( g(0) = 3 \), the y-intercept is \( 3 \).
- The values of \( g(x) \) are decreasing, but we don't have information about \( g(x) \) for \( x > 1 \) to determine its end behavior.
3. **Interceptions**:
- Both functions have the same x-intercept at \( x = 1 \).
- The y-intercepts are different: \( f \) has a y-intercept of \( 6 \) while \( g \) has a y-intercept of \( 3 \).
4. **End Behavior**:
- As \( x \) approaches \( ∞\):
- The function \( f \) (being an exponential function) will tend to \( ∞\).
- The behavior of \( g(x) \) as \( x \to ∞ \) cannot be determined from the table alone. We only have values for \( x = -1, 0, 1, 2, 3 \).
Based on this analysis, we can conclude:
- **Statement A** is incorrect because they have different y-intercepts.
- **Statement B** is correct in that they have the same x-intercept but we cannot conclude that their end behavior is different due to lack of information about \( g(x) \) as \( x \to ∞\).
- **Statement C** is incorrect as discussed above.
- **Statement D** is incorrect because they do not have the same y-intercept.
Thus, none of the statements perfectly describe the comparison, but the closest is **B**, assuming we assert that \( g \) behaves differently in end behavior (since we do not have sufficient information on that front). So, the answer is:
**B. They have the same x-intercept but different end behavior as x approaches ∞.**