The graph of f(x)=√x is reflected across the y-axis to create the graph of function g. How do the domains of ƒ and g compare?

A. The domains of f and g are both x≥ 0.
B. The domains of f and g are both all real numbers.
C. The domain of f is x≥ 0, while the domain of g is x≤ 0.
D. The domain of f is x≤ 0, while the domain of g is x≥ 0.

The domain of f ( x ) = x ​ is x ≥ 0 .
Reflecting f ( x ) across the y-axis gives g ( x ) = − x ​ .
The domain of g ( x ) = − x ​ is x ≤ 0 .
Therefore, the domain of f is x ≥ 0 , while the domain of g is x ≤ 0 .
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Explanation

Domain of f(x) The function f ( x ) = x ​ is defined only for non-negative values of x , since the square root of a negative number is not a real number. Therefore, the domain of f ( x ) is x ≥ 0 .

Reflection across y-axis When a function f ( x ) is reflected across the y-axis, the resulting function is g ( x ) = f ( − x ) . In this case, g ( x ) = − x ​ .

Domain of g(x) The function g ( x ) = − x ​ is defined only when − x ≥ 0 . Multiplying both sides of the inequality by − 1 (and flipping the inequality sign) gives x ≤ 0 . Therefore, the domain of g ( x ) is x ≤ 0 .

Comparison of Domains Comparing the domains of f ( x ) and g ( x ) , we see that the domain of f ( x ) is x ≥ 0 , while the domain of g ( x ) is x ≤ 0 .


Examples
Understanding the domains of functions and how they change with transformations like reflections is crucial in many real-world applications. For example, consider a scenario where f ( x ) = x ​ represents the distance a car can travel based on the amount of fuel x in its tank. Reflecting this function across the y-axis to get g ( x ) = − x ​ might seem abstract, but it could represent a situation where the car's fuel consumption is measured in reverse, perhaps due to a change in the measurement system or a theoretical model exploring reversed fuel flow. Knowing the domain of each function helps us understand the valid inputs for these models, ensuring we only consider realistic scenarios (e.g., non-negative fuel amounts).

Answered by GinnyAnswer | 2025-07-05

The domain of the function f ( x ) = x ​ is x ≥ 0 , while the domain of the reflected function g ( x ) = − x ​ is x ≤ 0 . Therefore, the correct option is C: The domain of f is x ≥ 0 , while the domain of g is x ≤ 0 .
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Answered by Anonymous | 2025-07-07