What is the multiplicative rate of change for the exponential function f(x)=2((5)/(2))^(-x)
Which of these following statements are true?
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The Correct Answer and Explanation is:
Let’s analyze and solve the question step by step.
Part A: Multiplicative Rate of Change
The multiplicative rate of change of an exponential function is the base of the exponent (what the function is multiplied by each time x increases by 1).
Given the function: f(x)=2(52)−xf(x) = 2\left(\frac{5}{2}\right)^{-x}
We need to identify the base of the exponential expression.
Step 1: Understand the exponent
(52)−x=(25)x\left(\frac{5}{2}\right)^{-x} = \left(\frac{2}{5}\right)^x
This is because a−x=1axa^{-x} = \frac{1}{a^x}. So: f(x)=2⋅(25)xf(x) = 2 \cdot \left(\frac{2}{5}\right)^x
Now, we can see that the base (or the multiplicative rate of change) is: 25\boxed{\frac{2}{5}}
This means that for every increase of 1 in xx, the output of the function is multiplied by 25\frac{2}{5}.
Part B: Which of the following statements are true?
Since options (a) and (b) are not complete in your message, I’ll assume you meant to list some statements related to the function. Let me provide two common statements based on typical exponential function analysis:
Statement (a): The function is increasing.
Statement (b): The function is decreasing.
To determine which is true, consider the form again: f(x)=2(25)xf(x) = 2\left(\frac{2}{5}\right)^x
The base 25\frac{2}{5} is less than 1, so this is a decreasing exponential function. As xx increases, f(x)f(x) gets smaller.
Thus, Statement (b) is true, and Statement (a) is false.
✅ Final Answers:
A. The multiplicative rate of change is 25\boxed{\frac{2}{5}}.
B. (b) is true\boxed{\text{(b) is true}}
📘 Explanation (300+ words):
Exponential functions have the general form f(x)=abxf(x) = ab^x, where:
- aa is the initial value or y-intercept,
- bb is the base, also called the multiplicative rate of change,
- If b>1b > 1, the function is increasing,
- If 0<b<10 < b < 1, the function is decreasing.
In the problem, we’re given: f(x)=2(52)−xf(x) = 2\left(\frac{5}{2}\right)^{-x}
We start by simplifying this expression. Recall that a negative exponent means taking the reciprocal: (52)−x=(25)x\left(\frac{5}{2}\right)^{-x} = \left(\frac{2}{5}\right)^x
Therefore, we rewrite the function as: f(x)=2(25)xf(x) = 2\left(\frac{2}{5}\right)^x
Now, the base is clearly 25\frac{2}{5}, and that is the multiplicative rate of change. This tells us that the output of the function is multiplied by 25\frac{2}{5} every time x increases by 1. Since 25\frac{2}{5} is a value between 0 and 1, this function is decreasing. That means as x gets larger, f(x)f(x) gets smaller.
In part B, we’re asked which statement is true between (a) and (b), and although their content was missing, a common pair of choices involves identifying whether a function is increasing or decreasing. Because our base is 25\frac{2}{5}, and that’s less than 1, we know the function is decreasing. Therefore, the true statement is (b): “The function is decreasing.”
Understanding how the base of an exponential function controls the direction of the graph is critical in algebra and precalculus. It helps in interpreting growth and decay models in science, finance, and real-life situations.