Question
Consider the two exponential functions.
Function 1
f (x) = 140(6)*
Function 2
-1
1
2
3
g (x)
42.5
170
680
2, 720
10, 880
Which two statements are true?
A As x increases, f(x) increases by a greater percentage than g(x).
B As x increases, f (x) increases by a lesser percentage than g (x).
C As x increases, f (x) increases by the same percentage as g (x).
D The y-intercept of f (x) is greater than the y-intercept of g(x).
E The y-intercept of f(x) is less than the y-intercept of
Function 1
f (x) = 140(6)*
Function 2
-1
1
2
3
g (x)
42.5
170
680
2, 720
10, 880
Which two statements are true?
A As x increases, f(x) increases by a greater percentage than g(x).
B As x increases, f (x) increases by a lesser percentage than g (x).
C As x increases, f (x) increases by the same percentage as g (x).
D The y-intercept of f (x) is greater than the y-intercept of g(x).
E The y-intercept of f(x) is less than the y-intercept of
Answers
GPT-4o mini
To evaluate the two exponential functions and determine which statements are true, let's first analyze each function.
Starting with Function 1:
\[ f(x) = 140(6^x) \]
This function starts at \( f(0) = 140 \times 6^0 = 140 \times 1 = 140 \).
Now, let's look at Function 2 based on the \( g(x) \) values provided:
- \( g(0) = 42.5 \)
- \( g(1) = 170 \)
- \( g(2) = 680 \)
- \( g(3) = 2,720 \)
- \( g(4) = 10,880 \)
Now let's find the y-intercept for function \( g(x) \):
Since we have values at \( g(0) = 42.5 \), the y-intercept of \( g(x) \) is 42.5.
Now we can check the statements one by one:
A. As x increases, \( f(x) \) increases by a greater percentage than \( g(x) \).
- To check this, we can calculate the percentage increase for both functions at a couple of points.
- For \( f(x) \), the percentage increase can be calculated between successive values. Similarly, calculate the percentage increase for \( g(x) \).
- Given that both functions are exponential, it is generally true that \( f(x) \) will increase at a higher rate than \( g(x) \) due to the base of the exponential in \( f(x) \) being higher. This statement is likely **true**.
B. As x increases, \( f(x) \) increases by a lesser percentage than \( g(x) \).
- This is the opposite of statement A and is likely **false**.
C. As x increases, \( f(x) \) increases by the same percentage as \( g(x) \).
- Because the bases of the exponentials are different, this statement is **false**.
D. The y-intercept of \( f(x) \) is greater than the y-intercept of \( g(x) \).
- The y-intercept of \( f(x) \) is 140, which is greater than the y-intercept of \( g(x) \) (42.5). This statement is **true**.
E. The y-intercept of \( f(x) \) is less than the y-intercept of \( g(x) \).
- Since 140 is greater than 42.5, this statement is **false**.
In conclusion, the two statements that are true are:
- A: As x increases, \( f(x) \) increases by a greater percentage than \( g(x) \).
- D: The y-intercept of \( f(x) \) is greater than the y-intercept of \( g(x) \).
Starting with Function 1:
\[ f(x) = 140(6^x) \]
This function starts at \( f(0) = 140 \times 6^0 = 140 \times 1 = 140 \).
Now, let's look at Function 2 based on the \( g(x) \) values provided:
- \( g(0) = 42.5 \)
- \( g(1) = 170 \)
- \( g(2) = 680 \)
- \( g(3) = 2,720 \)
- \( g(4) = 10,880 \)
Now let's find the y-intercept for function \( g(x) \):
Since we have values at \( g(0) = 42.5 \), the y-intercept of \( g(x) \) is 42.5.
Now we can check the statements one by one:
A. As x increases, \( f(x) \) increases by a greater percentage than \( g(x) \).
- To check this, we can calculate the percentage increase for both functions at a couple of points.
- For \( f(x) \), the percentage increase can be calculated between successive values. Similarly, calculate the percentage increase for \( g(x) \).
- Given that both functions are exponential, it is generally true that \( f(x) \) will increase at a higher rate than \( g(x) \) due to the base of the exponential in \( f(x) \) being higher. This statement is likely **true**.
B. As x increases, \( f(x) \) increases by a lesser percentage than \( g(x) \).
- This is the opposite of statement A and is likely **false**.
C. As x increases, \( f(x) \) increases by the same percentage as \( g(x) \).
- Because the bases of the exponentials are different, this statement is **false**.
D. The y-intercept of \( f(x) \) is greater than the y-intercept of \( g(x) \).
- The y-intercept of \( f(x) \) is 140, which is greater than the y-intercept of \( g(x) \) (42.5). This statement is **true**.
E. The y-intercept of \( f(x) \) is less than the y-intercept of \( g(x) \).
- Since 140 is greater than 42.5, this statement is **false**.
In conclusion, the two statements that are true are:
- A: As x increases, \( f(x) \) increases by a greater percentage than \( g(x) \).
- D: The y-intercept of \( f(x) \) is greater than the y-intercept of \( g(x) \).