The square root function, written as $f(x) = \sqrt{x}$, introduces a unique challenge when determining its domain and range. The domain consists of all possible input values (x-values) that produce a real number output. Since we cannot take the square root of a negative number and get a real result, the domain is restricted to non-negative numbers. The range consists of all possible output values (y-values) that the function can produce.
Understanding transformations of the basic square root function, such as shifts and reflections, is key to finding the domain and range of more complex square root functions. For example, in the function $f(x) = a\sqrt{x-h} + k$, 'h' shifts the graph horizontally, affecting the domain, while 'k' shifts the graph vertically, affecting the range. The value of 'a' determines whether the graph is reflected and impacts the range as well.
Match the following terms with their definitions:
| Term | Definition |
|---|---|
| 1. Domain | A. The set of all possible output values of a function. |
| 2. Range | B. A function that contains a square root of a variable. |
| 3. Square Root Function | C. A transformation that flips a graph over a line. |
| 4. Transformation | D. The set of all possible input values of a function. |
| 5. Reflection | E. Alteration of a function's graph by shifting, stretching, or reflecting. |
The domain of a square root function $f(x) = \sqrt{x-a}$ is all real numbers greater than or equal to _____. The range of $f(x) = \sqrt{x} + b$ is all real numbers greater than or equal to _____. A _____ shifts the square root function horizontally, while a vertical shift changes the _____. The basic square root function, $f(x) = \sqrt{x}$, has a domain of $[0, \infty)$ and a range of _____. If there is a negative sign in front of the square root, it creates a _____ across the x-axis.
Explain how changing the value of 'a' in the function $f(x) = \sqrt{x} + a$ affects the graph of the function. How does this change impact the domain and the range?
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