Math Problem Statement

Given the functions f(x) = 3^x and g(x) = (1/3)^x, analyze the behavior of these exponential functions.

Solution

The image shows two functions:

  1. f(x)=3xf(x) = 3^x
  2. g(x)=(13)xg(x) = \left(\frac{1}{3}\right)^x

These are exponential functions with different bases.

Explanation of Each Function:

  1. f(x)=3xf(x) = 3^x:

    • This is an exponential growth function because the base, 33, is greater than 1.
    • As xx increases, f(x)f(x) grows rapidly.
  2. g(x)=(13)xg(x) = \left(\frac{1}{3}\right)^x:

    • This is an exponential decay function because the base, 13\frac{1}{3}, is between 0 and 1.
    • As xx increases, g(x)g(x) decreases, approaching zero.

If you'd like, I can calculate specific values, compare these functions, or explain their properties further.

Would you like more details or have any questions?

Related Questions:

  1. What are the properties of exponential growth and decay functions?
  2. How do the graphs of f(x)=3xf(x) = 3^x and g(x)=(13)xg(x) = \left(\frac{1}{3}\right)^x compare?
  3. What is the effect of the base in exponential functions on their behavior?
  4. How would these functions behave as xx \to -\infty and x+x \to +\infty?
  5. What transformations can be applied to these functions, and how do they affect their graphs?

Tip:

For exponential functions axa^x where a>1a > 1, the function represents growth. For 0<a<10 < a < 1, it represents decay.

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Growth and Decay
Function Analysis

Formulas

f(x) = a^x
g(x) = (1/a)^x for a > 1

Theorems

Exponential Growth and Decay Properties

Suitable Grade Level

Grades 9-11