Evaluate (f/g) (x) if f (x)= √x-2 and g (x) = 1/x

To evaluate \(\left(\frac{f}{g}\right)(x)\) given that \(f(x) = \sqrt{x-2}\) and \(g(x) = \frac{1}{x}\), follow these steps:

### Step 1: Write the expression for \(\left(\frac{f}{g}\right)(x)\)
\[
\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}
\]

### Step 2: Substitute the given functions \(f(x)\) and \(g(x)\) into the expression
\[
\left(\frac{f}{g}\right)(x) = \frac{\sqrt{x-2}}{\frac{1}{x}}
\]

### Step 3: Simplify the expression
Dividing by \(\frac{1}{x}\) is equivalent to multiplying by \(x\):
\[
\left(\frac{f}{g}\right)(x) = \sqrt{x-2} \times x
\]

### Step 4: Write the final answer
\[
\left(\frac{f}{g}\right)(x) = x\sqrt{x-2}
\]

This is the expression for \(\left(\frac{f}{g}\right)(x)\).

Would you like more details or have any questions? Here are some related problems you might find interesting:

1. Find \(\left(\frac{f \cdot g}{h}\right)(x)\) if \(f(x) = \sqrt{x-2}\), \(g(x) = \frac{1}{x}\), and \(h(x) = x^2\).
2. Evaluate \(\left(f \cdot g\right)(x)\) for the same functions \(f(x)\) and \(g(x)\).
3. Determine the domain of \(\left(\frac{f}{g}\right)(x)\).
4. Simplify \(\left(\frac{f}{h}\right)(x)\) if \(h(x) = x+2\) and \(f(x) = \sqrt{x-2}\).
5. Find the inverse of the function \(f(x) = x\sqrt{x-2}\).

**Tip:** When dividing by a function, always ensure that the denominator is not zero to avoid undefined values.

Learn how to evaluate the rational function (f/g)(x) given f(x) = square root of (x-2) and g(x) = 1/x. Follow a step-by-step solution to simplify and find the expression. Suitable for advanced high school students.

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