Math Problem Statement

The query submitted by the user involves analyzing the functions f(x) = 6^x, f(x) = 5^x, f(x) = 6^{-x}, and f(x) = 5^{-x} and constructing a table of values for specific X values.

Solution

The image you uploaded shows a blackboard with four functions and a table of values labeled XX and F(X)F(X). The functions written on the board are:

  1. f(x)=6xf(x) = 6^x
  2. f(x)=5xf(x) = 5^x
  3. f(x)=6xf(x) = 6^{-x}
  4. f(x)=5xf(x) = 5^{-x}

There is also a table of values for XX and F(X)F(X) with the following entries:

X & F(X) \\ \hline 3 & \\ 2 & \\ 1 & \\ 0 & \\ -1 & \\ -2 & \\ \end{array}$$ It seems like the task is to fill in the table for $$F(X)$$ based on the provided functions. If you'd like, I can help you calculate the values of $$F(X)$$ for each function at the specified $$X$$ values. Would you like to proceed with that, or do you have any specific questions about the functions or table? --- **Relative Questions:** 1. How does the behavior of exponential functions differ when the base is greater than 1 versus between 0 and 1? 2. What effect does negating the exponent in an exponential function have on the graph of the function? 3. How would you describe the asymptotes of the functions $$f(x) = 6^x$$ and $$f(x) = 6^{-x}$$? 4. Can you explain the process of constructing a table of values for any function, given specific $$X$$ values? 5. How does the function $$f(x) = 5^x$$ compare to $$f(x) = 6^x$$ in terms of growth rate? **Tip:** When working with exponential functions, it's essential to understand how the base and the sign of the exponent affect the function's growth or decay.

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Function Evaluation
Graphing Functions

Formulas

f(x) = a^x
f(x) = a^{-x}

Theorems

Properties of Exponential Functions
Behavior of Exponential Growth and Decay

Suitable Grade Level

Grades 9-12