Understanding Exponential Functions: Properties, Domains, Ranges, and Transformations

Exponential Function

An exponential function is a mathematical function of the form: f(x) = a^x, where ‘a’ is a positive constant called the base, and ‘x’ is the exponent

An exponential function is a mathematical function of the form: f(x) = a^x, where ‘a’ is a positive constant called the base, and ‘x’ is the exponent. The exponent can be any real number.

Here are some key properties of exponential functions:

1. Growth and Decay:
– If 0 < a < 1, the function exhibits exponential decay. As x increases, the function approaches zero. - If a > 1, the function exhibits exponential growth. As x increases, the function increases rapidly.

2. Domain and Range:
– The domain of an exponential function is all real numbers.
– The range of an exponential function depends on the base.
– If 0 < a < 1, the range is (0, ∞). - If a > 1, the range is (0, ∞) if a is positive, and (-∞, 0) if a is negative.

3. Intercepts:
– The x-intercept is the value of x when f(x) = 0. For an exponential function, this occurs when x = -∞ (for decay) or when x = ∞ (for growth).
– The y-intercept is the value of f(x) when x = 0. This is equal to 1, since any non-zero constant raised to the power of zero is 1.

4. Asymptotes:
– If 0 < a < 1, the horizontal asymptote of the graph is the x-axis (y = 0). - If a > 1, the horizontal asymptote of the graph is the y-axis (x = 0).

5. Transformations:
– Exponential functions can undergo various transformations such as translations, reflections, and dilations.
– A translation by (h, k) shifts the graph horizontally by ‘h’ units and vertically by ‘k’ units.
– A reflection across the x-axis changes the sign of the base ‘a’, causing the graph to be reflected.
– A dilation by a factor of ‘c’ stretches or compresses the graph vertically, changing the base ‘a’ to a^c.

These are the general concepts and properties related to exponential functions. If you have any specific questions or need further clarification on any topic, feel free to ask!

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