1. What is the domain of the function [tex]$f(x)=\sqrt{x}$[/tex]?
2. What is the range of [tex]$f(x)=e^x$[/tex]?
Answer (2)
The domain of f ( x ) = x is found by recognizing that the square root function is only defined for non-negative numbers: x ≥ 0 .
The range of f ( x ) = e x is determined by understanding that the exponential function always yields positive values: 0"> y > 0 .
Therefore, the domain of f ( x ) = x is x ≥ 0 and the range of f ( x ) = e x is 0}"> y > 0 .
Explanation
Problem Analysis We are asked to find the domain of the function f ( x ) = x and the range of the function f ( x ) = e x .
Domain of f ( x ) = x The domain of a function is the set of all possible input values (x-values) for which the function is defined. For f ( x ) = x , the square root is only defined for non-negative values. Therefore, the domain is all x such that x ≥ 0 .
Range of f ( x ) = e x The range of a function is the set of all possible output values (y-values) that the function can produce. For f ( x ) = e x , the exponential function is always positive for any real number x . As x approaches negative infinity, e x approaches 0, but never actually reaches it. As x approaches positive infinity, e x also approaches positive infinity. Therefore, the range is all y such that 0"> y > 0 .
Final Answer The domain of f ( x ) = x is x ≥ 0 , and the range of f ( x ) = e x is 0"> y > 0 .
Examples
Understanding domains and ranges is crucial in many real-world applications. For example, when modeling population growth with an exponential function, the domain represents time (which can't be negative), and the range represents the population size (which must be positive). Similarly, in physics, when dealing with distances or velocities, understanding the domain and range ensures that the mathematical model aligns with physical reality, preventing nonsensical results like negative distances.
The domain of the function f ( x ) = x is x ≥ 0 , meaning it only accepts non-negative inputs. The range of the function f ( x ) = e x is 0"> y > 0 , which signifies it can only produce positive outputs. This shows the behavior and limits of these two mathematical functions in terms of acceptable inputs and outputs.
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