Math Problem Statement
Based on the provided table of x and f(x) values: x = {1, 2, 3, 4} and f(x) = {8, 6.4, 5.12, 4.096}, identify the function that models the data.
Solution
The table provided displays values for and . Based on the values:
\hline x & f(x) \\ \hline 1 & 8 \\ 2 & 6.4 \\ 3 & 5.12 \\ 4 & 4.096 \\ \hline \end{array}$$ It appears that $$f(x)$$ is decreasing as $$x$$ increases, suggesting exponential decay. Let me analyze further to derive a potential function for $$f(x)$$. ### Step 1: Check for exponential relationship The general form of an exponential function is: $$f(x) = a \cdot b^x$$ where $$a$$ is the initial value and $$b$$ is the base. 1. At $$x = 1$$, $$f(1) = 8$$, so $$a \cdot b^1 = 8$$. 2. At $$x = 2$$, $$f(2) = 6.4$$, so $$a \cdot b^2 = 6.4$$. ### Step 2: Solve for $$b$$ Dividing the equation for $$f(2)$$ by $$f(1)$$: $$\frac{a \cdot b^2}{a \cdot b^1} = \frac{6.4}{8}$$ $$b = \frac{6.4}{8} = 0.8$$ ### Step 3: Solve for $$a$$ From $$f(1) = 8$$: $$a \cdot 0.8^1 = 8 \implies a = 8 / 0.8 = 10$$ ### Final Function: The function is: $$f(x) = 10 \cdot (0.8)^x$$ ### Verification: - For $$x = 1$$: $$f(1) = 10 \cdot 0.8 = 8$$ ✓ - For $$x = 2$$: $$f(2) = 10 \cdot 0.8^2 = 10 \cdot 0.64 = 6.4$$ ✓ - For $$x = 3$$: $$f(3) = 10 \cdot 0.8^3 = 10 \cdot 0.512 = 5.12$$ ✓ - For $$x = 4$$: $$f(4) = 10 \cdot 0.8^4 = 10 \cdot 0.4096 = 4.096$$ ✓ The function accurately models the data. --- Do you want further details on how exponential functions work or related concepts? ### Related Questions: 1. What is the difference between exponential growth and decay? 2. How can I determine the decay rate of an exponential function from data? 3. Can exponential decay be applied to real-life scenarios (e.g., radioactive decay)? 4. How would the function change if $$b > 1$$? 5. What is the significance of the constant $$a$$ in exponential functions? ### Tip: When identifying exponential functions from data, divide consecutive outputs to calculate the base $$b$$, then work backward to determine $$a$$.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Exponential Functions
Exponential Decay
Formulas
f(x) = a * b^x
Theorems
Exponential Decay
Suitable Grade Level
Grades 9-12