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:
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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.
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Asymptote: The x-axis (y = 0) serves as a horizontal asymptote for the function.
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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'.
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Zeros and Y-intercept: The function has no zeros (roots), and its y-intercept is always 1.
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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.
- 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.
