Understanding the Domain and Range of f(x) = 43√(81x)
To determine the domain and range of the function f(x) = 43√(81x), we need to understand the constraints of the function based on its mathematical expression.
Domain
The domain of a function represents all possible input values (x-values) that can be used in the function without causing any mathematical inconsistencies.
In this case, we have a square root function. For square roots, the expression inside must be non-negative:
81x ≥ 0
To find the domain, we solve this inequality:
x ≥ 0
Therefore, the domain of the function f(x) = 43√(81x) is:
[0, ∞)
Range
The range of a function includes all possible output values (y-values) that the function can produce. For the function f(x) = 43√(81x), we need to consider what happens as x varies within the domain:
– When x = 0, f(0) = 43√(81 * 0) = 43 * 0 = 0.
– As x increases, √(81x) will also increase, leading to larger values for f(x).
Since the square root function produces non-negative results and we multiply by 43, the smallest value of f(x) is 0 (when x = 0), and it can grow infinitely as x increases.
Therefore, the range of the function f(x) = 43√(81x) is:
[0, ∞)
Summary
In conclusion, the domain and range of the function f(x) = 43√(81x) are:
- Domain: [0, ∞)
- Range: [0, ∞)