Verified Answer :
- The domain of the function is found by setting the radicand \(x-5 \ge 0\), resulting in \(x \ge 5\).
- The range is determined by the output of the square root, which is always \(\ge 0\), resulting in \(y \ge 0\).
- The x-intercept is calculated by solving \(2\sqrt{x - 5} = 0\), which gives \(x = 5\).
- There is no y-intercept because \(x=0\) is not in the domain.
- The properties are: \(\boxed{\text{Domain: } [5, \infty); \text{ Range: } [0, \infty); \text{ x-intercept: } (5, 0); \text{ No y-intercept}}\)
Explanation
- Analyzing the Function h(x) = 2\sqrt{x - 5}
The function \(h(x) = 2\sqrt{x - 5}\) is a transformation of the parent square root function \(f(x) = \sqrt{x}\). To analyze its properties, we consider the definition of the square root and the effect of the transformations. The square root function \(\sqrt{u}\) is defined only for \(u \ge 0\), and its output is always non-negative. This foundational knowledge allows us to determine the domain, range, and other key features of the function.
- Determining Domain and Range
To find the domain, we must ensure the expression inside the square root is non-negative:
\[x - 5 \ge 0 \implies x \ge 5.\]
Thus, the domain is \([5, \infty)\). Regarding the range, since \(\sqrt{x - 5} \ge 0\), the product \(2\sqrt{x - 5}\) is also at least 0. As \(x\) approaches infinity, \(h(x)\) also increases without bound. Therefore, the range is \([0, \infty)\).
- Intercepts and Graph Transformations
The x-intercept occurs where \(h(x) = 0\):
\[2\sqrt{x - 5} = 0 \implies \sqrt{x - 5} = 0 \implies x - 5 = 0 \implies x = 5.\]
The x-intercept is the point \((5, 0)\).
The y-intercept occurs at \(x = 0\). However, \(0\) is not in the domain (\(x \ge 5\)), so the function does not have a y-intercept. In terms of transformations, the graph of \(h(x)\) is the graph of \(f(x) = \sqrt{x}\) shifted to the right by 5 units and vertically stretched by a factor of 2.
- Conclusion and Summary of Properties
In summary, the function \(h(x) = 2\sqrt{x - 5}\) has a domain of \(x \ge 5\), a range of \(y \ge 0\), an x-intercept at \(5\), and no y-intercept. It represents a horizontal shift of the standard square root function. The final answer describing these properties is \(\boxed{\text{Domain: } [5, \infty); \text{ Range: } [0, \infty); \text{ x-intercept: } (5, 0); \text{ No y-intercept}}\).
Examples
Understanding the properties of functions like \(h(x) = 2\sqrt{x - 5}\) is essential in fields like physics and engineering. For example, if this function modeled the velocity of a particle starting at a position \(x=5\), the domain would represent the physical space where the particle exists, and the range would describe the possible velocities it can achieve. Recognizing domain and range helps in predicting the behavior of such systems and ensuring calculations remain within physically meaningful bounds.