The domain of an exponential function is all real numbers, while the range depends on the function's specific form and any transformations applied.
Understanding Domain and Range
- Domain: The set of all possible input values (x-values) for which the function is defined.
- Range: The set of all possible output values (y-values) that the function can produce. The range of a function is the set of output values for the dependent variable.
Domain of Exponential Functions
For any exponential function of the form f(x) = abx, where 'a' and 'b' are constants and 'b' is greater than 0 and not equal to 1, the domain is always all real numbers. This means you can plug in any real number for 'x', and the function will produce a valid output.
Range of Exponential Functions
The range is more nuanced and depends on the form of the exponential function:
- Basic Exponential Function (f(x) = abx, where a > 0): The range is all positive real numbers, often expressed as (0, ∞). This is because bx is always positive, and multiplying by a positive 'a' keeps it positive.
- Exponential Function with Vertical Shift (f(x) = abx + k): The range is (k, ∞) if a > 0, and (-∞, k) if a < 0. 'k' represents a vertical shift of the horizontal asymptote. The range, however, is bounded by the horizontal asymptote of the graph of f(x).
- Exponential Function with Reflection (f(x) = -abx): If 'a' is negative, the function is reflected over the x-axis. The range becomes (-∞, 0).
Examples
| Function | Domain | Range |
|---|---|---|
| f(x) = 2x | (-∞, ∞) | (0, ∞) |
| f(x) = 3 * 2x | (-∞, ∞) | (0, ∞) |
| f(x) = 2x + 1 | (-∞, ∞) | (1, ∞) |
| f(x) = -2x | (-∞, ∞) | (-∞, 0) |
| f(x) = -2x + 5 | (-∞, ∞) | (-∞, 5) |
Key Takeaways
- The domain of f(x) = abx is always all real numbers.
- The range depends on the value of 'a' and any vertical shifts. It's bounded by the horizontal asymptote.
- If 'a' is positive and there's no vertical shift, the range is (0, ∞).
- Vertical shifts (adding or subtracting a constant) will shift the range accordingly.
- A negative 'a' reflects the function over the x-axis, impacting the range.
