Math Problem Statement
Consider the two exponential functions Function A: y = 2^x + 3 and Function B (values provided in a table), and select the true statement about their behavior as x approaches positive or negative infinity.
Solution
Let's analyze the problem step by step.
Function A:
Function A is given as:
This is an exponential function where the base is 2, and it has a vertical shift of 3. As , grows very large, and so does . As , approaches 0, making approach 3.
Function B:
We have the values of for Function B in a table:
\hline x & y \\ \hline 0 & 2 \\ 1 & 8 \\ 2 & 28 \\ 3 & 26 \\ 4 & 80 \\ \hline \end{array}$$ Looking at the values of $$y$$, it seems that Function B grows quite quickly as $$x$$ increases, suggesting it might be a polynomial or exponential function. ### Evaluating the Answer Choices: - **A**: As $$x \to -\infty$$, Function A approaches $$y = 3$$, but we don't have enough information to say if Function B behaves similarly without knowing its explicit formula. So, this statement can't be assumed as true. - **B**: This statement suggests that as $$x \to -\infty$$, the value of Function B exceeds that of Function A. But we know that Function A approaches a positive constant (3), and exponential functions generally tend to decrease as $$x$$ decreases to negative infinity. Without an explicit expression for Function B, this statement is unlikely to be true. - **C**: As $$x \to \infty$$, $$2^x$$ grows exponentially, so Function A will also grow exponentially. Based on the table, Function B seems to grow quickly, but since Function A is exponential and the table only shows values for small $$x$$, we cannot assume that Function B will always be larger than Function A. This statement may not be true either. - **D**: This statement is likely true because exponential functions like $$2^x + 3$$ grow much faster than most polynomial or even other exponential functions after a certain point. Therefore, for sufficiently large $$x$$, Function A will likely exceed Function B, which aligns with the general behavior of exponential growth. ### Conclusion: The most likely **correct answer** is **D**, as exponential functions typically outgrow most other functions as $$x \to \infty$$. Would you like more detailed insights into this, or do you have any other questions? Here are 5 related questions: 1. What happens to exponential functions as $$x \to -\infty$$? 2. How do you determine the growth rate of a function based on its equation or table? 3. How does the addition of a constant affect the graph of an exponential function? 4. What are the key differences between polynomial and exponential growth? 5. Can you derive the formula for Function B from the table? **Tip**: In exponential functions, pay attention to whether the exponent is positive or negative—it greatly affects the growth behavior of the function!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Exponential Functions
Limits
Formulas
y = 2^x + 3 (Function A)
Table values for Function B
Theorems
Exponential growth
Suitable Grade Level
Grades 9-12
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