To determine the domain and range of the function \( d(x) = 5(x - 9)^2 - 32 \), let's analyze it step by step.
Domain
The function d(x) is a polynomial function (specifically a quadratic function) defined for all real numbers. Therefore, the domain of the function is:
Domain: \( { x \in \mathbb{R} } \) or simply \( \mathbb{R} \).
Range
To find the range, we can analyze the quadratic function \( d(x) = 5(x - 9)^2 - 32 \).
- The coefficient of the squared term is positive (5), which means the parabola opens upwards.
- The vertex form of the function is already given: \( 5(x - 9)^2 - 32 \).
- The vertex is at the point (9, -32).
- Since the parabola opens upwards and the vertex represents the minimum point, the minimum value of the function is -32.
Thus, the range of the function is all values greater than or equal to -32:
Range: \( { y \in \mathbb{R} | y \geq -32 } \).
Conclusion
Putting it all together:
- Domain: \( { x \in \mathbb{R} } \)
- Range: \( { y \in \mathbb{R} | y \geq -32 } \)
From the provided options, the correct answer that reflects these findings is:
d) Domain = \( { x \in \mathbb{R} } \)
Range = \( { y \in \mathbb{R} | y \geq -32 } \)