Question
A table representing the function f(x) = 2(three-halves) Superscript x is shown below.A 2-column table has 4 rows. The first column is labeled x with entries (0, 1, 2, 3). The second column is labeled f (x) with entries 2, 3, 4.5, 6.75.
What is true of the given function?
The function increases at a constant additive rate.
The function increases at a constant multiplicative rate.
The function has an initial value of 0.
As each x value increases by 1, the y values increase by 1.
What is true of the given function?
The function increases at a constant additive rate.
The function increases at a constant multiplicative rate.
The function has an initial value of 0.
As each x value increases by 1, the y values increase by 1.
Answers
GPT-4o mini
To analyze the function \( f(x) = 2\left(\frac{3}{2}\right)^{x} \), let's examine each statement based on the provided table.
1. **The function increases at a constant additive rate.**
- This statement is false because the increments in the \( f(x) \) values are not constant. The differences between consecutive \( f(x) \) values are not the same: \( 3 - 2 = 1 \), \( 4.5 - 3 = 1.5 \), and \( 6.75 - 4.5 = 2.25 \).
2. **The function increases at a constant multiplicative rate.**
- This statement is true. To see this, let’s check the ratio of consecutive \( f(x) \) values:
- \( \frac{f(1)}{f(0)} = \frac{3}{2} \)
- \( \frac{f(2)}{f(1)} = \frac{4.5}{3} = \frac{3}{2} \)
- \( \frac{f(3)}{f(2)} = \frac{6.75}{4.5} = \frac{3}{2} \)
Thus, the function increases by a factor of \( \frac{3}{2} \) each time \( x \) increases by 1.
3. **The function has an initial value of 0.**
- This statement is false. At \( x = 0 \), \( f(0) = 2\left(\frac{3}{2}\right)^{0} = 2 \), so the initial value is 2, not 0.
4. **As each x value increases by 1, the y values increase by 1.**
- This statement is false. The increase in \( f(x) \) does not correspond to an addition of 1 for each increase in \( x \); we already established that the increases are not consistent.
Thus, the correct statement about the given function is:
**The function increases at a constant multiplicative rate.**
1. **The function increases at a constant additive rate.**
- This statement is false because the increments in the \( f(x) \) values are not constant. The differences between consecutive \( f(x) \) values are not the same: \( 3 - 2 = 1 \), \( 4.5 - 3 = 1.5 \), and \( 6.75 - 4.5 = 2.25 \).
2. **The function increases at a constant multiplicative rate.**
- This statement is true. To see this, let’s check the ratio of consecutive \( f(x) \) values:
- \( \frac{f(1)}{f(0)} = \frac{3}{2} \)
- \( \frac{f(2)}{f(1)} = \frac{4.5}{3} = \frac{3}{2} \)
- \( \frac{f(3)}{f(2)} = \frac{6.75}{4.5} = \frac{3}{2} \)
Thus, the function increases by a factor of \( \frac{3}{2} \) each time \( x \) increases by 1.
3. **The function has an initial value of 0.**
- This statement is false. At \( x = 0 \), \( f(0) = 2\left(\frac{3}{2}\right)^{0} = 2 \), so the initial value is 2, not 0.
4. **As each x value increases by 1, the y values increase by 1.**
- This statement is false. The increase in \( f(x) \) does not correspond to an addition of 1 for each increase in \( x \); we already established that the increases are not consistent.
Thus, the correct statement about the given function is:
**The function increases at a constant multiplicative rate.**
On a coordinate plane, an exponential function has a horizontal asymptote of y = 0. The function goes through points (negative 2, two-ninths), (negative 1, two-thirds), (0, 2), (1, 6), (2, 18), and (3, 54).What is the multiplicative rate of change for the exponential function graphed to the left?
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