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- and y-intercepts.
C.
They have the same y-intercept and the same end behavior as x approaches ∞.
D.
They have the same x-intercept but different 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- and y-intercepts.
C.
They have the same y-intercept and the same end behavior as x approaches ∞.
D.
They have the same x-intercept but different end behavior as x approaches ∞.
Answers
GPT-4o mini
To analyze the two functions \( f \) and \( g \), we can start by determining their intercepts and end behavior.
1. **Function \( f \)**:
- It passes through the points (−0.25, 10) and (0, 6), meaning:
- The y-intercept \( f(0) = 6 \).
- It also intercepts the x-axis at 1 unit, meaning:
- The x-intercept is at \( x = 1 \).
- Since \( f \) is an exponential function, as \( x \to \infty \), \( f(x) \) will approach either 0 (if the base of the exponential function is between 0 and 1) or infinity (if the base is greater than 1). However, since \( f(1) \) is 0, indicating it could potentially be a decreasing function.
2. **Function \( g \)**:
- From the table, we have the values:
- \( g(-1) = 15 \)
- \( g(0) = 3 \)
- \( g(1) = 0 \)
- This indicates that the y-intercept \( g(0) = 3 \) and the x-intercept \( g(1) = 0 \).
- The values provided suggest that \( g(x) \) is decreasing as \( x \) increases from -1 to 1.
Now, comparing \( f \) and \( g \):
- For **x-intercepts**:
- Both functions have the same x-intercept at \( x = 1 \).
- For **y-intercepts**:
- Function \( f \) has a y-intercept of \( 6 \) (at \( x = 0 \)).
- Function \( g \) has a y-intercept of \( 3 \) (at \( x = 0 \)).
- These are different.
- For **end behavior**:
- Function \( f \) (being an exponential function) will approach either 0 or infinity as \( x \to \infty \), depending on the base.
- Function \( g \) appears to be decreasing and approaches 0 (given that \( g(1) = 0 \), and the value \( g(0) = 3 \) is larger than \( g(1) = 0 \) suggests that as \( x \to \infty \), \( g(x) \) may continue to decrease, possibly approaching 0).
This leads us to the options:
A. They have different x- and y-intercepts but the same end behavior as \( x \) approaches \( \infty \). (False, different y-intercepts and unclear about end behavior similarity)
B. They have the same x- and y-intercepts. (False, different y-intercepts)
C. They have the same y-intercept and the same end behavior as \( x \) approaches \( \infty \). (False, different y-intercepts)
D. They have the same x-intercept but different end behavior as \( x \) approaches \( \infty \). (True, same x-intercept but different behaviors as explained)
The correct answer is **D**.
1. **Function \( f \)**:
- It passes through the points (−0.25, 10) and (0, 6), meaning:
- The y-intercept \( f(0) = 6 \).
- It also intercepts the x-axis at 1 unit, meaning:
- The x-intercept is at \( x = 1 \).
- Since \( f \) is an exponential function, as \( x \to \infty \), \( f(x) \) will approach either 0 (if the base of the exponential function is between 0 and 1) or infinity (if the base is greater than 1). However, since \( f(1) \) is 0, indicating it could potentially be a decreasing function.
2. **Function \( g \)**:
- From the table, we have the values:
- \( g(-1) = 15 \)
- \( g(0) = 3 \)
- \( g(1) = 0 \)
- This indicates that the y-intercept \( g(0) = 3 \) and the x-intercept \( g(1) = 0 \).
- The values provided suggest that \( g(x) \) is decreasing as \( x \) increases from -1 to 1.
Now, comparing \( f \) and \( g \):
- For **x-intercepts**:
- Both functions have the same x-intercept at \( x = 1 \).
- For **y-intercepts**:
- Function \( f \) has a y-intercept of \( 6 \) (at \( x = 0 \)).
- Function \( g \) has a y-intercept of \( 3 \) (at \( x = 0 \)).
- These are different.
- For **end behavior**:
- Function \( f \) (being an exponential function) will approach either 0 or infinity as \( x \to \infty \), depending on the base.
- Function \( g \) appears to be decreasing and approaches 0 (given that \( g(1) = 0 \), and the value \( g(0) = 3 \) is larger than \( g(1) = 0 \) suggests that as \( x \to \infty \), \( g(x) \) may continue to decrease, possibly approaching 0).
This leads us to the options:
A. They have different x- and y-intercepts but the same end behavior as \( x \) approaches \( \infty \). (False, different y-intercepts and unclear about end behavior similarity)
B. They have the same x- and y-intercepts. (False, different y-intercepts)
C. They have the same y-intercept and the same end behavior as \( x \) approaches \( \infty \). (False, different y-intercepts)
D. They have the same x-intercept but different end behavior as \( x \) approaches \( \infty \). (True, same x-intercept but different behaviors as explained)
The correct answer is **D**.