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The given function is: $G(x) = \frac{x\sqrt{x} - \sqrt{x}}{x - 1}$ Let's find the domain and range of this function.

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Domain and range of quadratic functions: equations

Full solution

Q. The given function is:

G(x) = \frac{x\sqrt{x} - \sqrt{x}}{x - 1}

Let's find the domain and range of this function.

Simplify $G(x)$ : First, let's simplify $G(x)$ to see if we can cancel out any terms. $\newline$ $G(x) = \frac{x\sqrt{x} - \sqrt{x}}{x - 1}$ $\newline$ Factor $\sqrt{x}$ out of the numerator. $\newline$ $G(x) = \frac{\sqrt{x}(x - 1)}{x - 1}$
Factor out $\sqrt{x}$ : Now, we cancel out the $(x - 1)$ terms. $\newline$ G(x) = $\sqrt{x}$ , for $x \neq 1$
Cancel out terms: To find the domain, we consider where the original function is defined. $\newline$ Since we cannot divide by zero, $x$ cannot be $1$ . $\newline$ Also, $\sqrt{x}$ is only real for $x \geq 0$ . $\newline$ So, the domain is $x \geq 0$ , $x \neq 1$ .
Find domain: For the range, we look at the simplified function $G(x) = \sqrt{x}$ . The square root function outputs values $y \geq 0$ . So, the range is $y \geq 0$ .

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