Exponential Function and Its Properties

This page provides a detailed examination of the exponential function and its key characteristics. The content is structured to give students a thorough understanding of this important mathematical concept.

The exponential function is defined as f(x) = a^x, where a > 0 and a ≠ 1. This definition is crucial for understanding the function's behavior and properties.

Definition: The exponential function is expressed as f(x) = a^x, where 'a' is the base and must be greater than 0 but not equal to 1.

Several important properties of the exponential function are highlighted:

1. Domain and Range: The function's domain is all real numbers (ℝ), while its range is (0, ∞), indicating that the function never touches or crosses the x-axis.

2. Asymptote: The x-axis (y = 0) serves as a horizontal asymptote for the function.

3. Intersection Point: Every exponential curve intersects the y-axis at the point (0, 1).

Highlight: A key characteristic of the exponential function is that it always passes through the point (0, 1), regardless of the base 'a'.

4. Zeros and Y-intercept: The function has no zeros (roots), and its y-intercept is always 1.

5. Monotonicity: The function's behavior depends on the value of 'a':

   • For a > 1, the function is increasing.
   • For 0 < a < 1, the function is decreasing.

Example: If a = 2, the function f(x) = 2^x is increasing, while if a = 1/2, the function f(x) = (1/2)^x is decreasing.

6. Symmetry: The graphs of y = a^x and y = (1/a)^x are symmetric with respect to the y-axis.

The page also includes a graphical representation of the exponential function, illustrating these properties visually. This combination of textual explanation and visual aid helps reinforce the understanding of the exponential function's behavior and characteristics.

Vocabulary:

• Dziedzina funkcji wykładniczej: The domain of the exponential function
• Zbiór wartości funkcji wykładniczej: The range of the exponential function
• Miejsce zerowe funkcji wykładniczej: The zero (root) of the exponential function (which does not exist)

This comprehensive overview provides students with a solid foundation for understanding the exponential function, its graph, and its fundamental properties, preparing them for more advanced topics and applications in mathematics.