given that F(x)=(x-1)/x and g(x)=1/x, obtain simplified expression for f=g, f-g, f*g and f/g.

Given the functions \( f(x) = \frac{x-1}{x} \) and \( g(x) = \frac{1}{x} \), let's obtain the simplified expressions for \( f + g \), \( f - g \), \( f \cdot g \), and \( \frac{f}{g} \).

### 1. **\( f + g \)**
\[ f(x) + g(x) = \frac{x-1}{x} + \frac{1}{x} \]

Since both fractions have the same denominator, we can combine them:
\[ f(x) + g(x) = \frac{(x-1) + 1}{x} \]

Simplifying the numerator:
\[ f(x) + g(x) = \frac{x - 1 + 1}{x} = \frac{x}{x} = 1 \]

So, the simplified expression is:
\[ \boxed{1} \]

### 2. **\( f - g \)**
\[ f(x) - g(x) = \frac{x-1}{x} - \frac{1}{x} \]

Again, both fractions have the same denominator, so we can subtract the numerators:
\[ f(x) - g(x) = \frac{(x-1) - 1}{x} \]

Simplifying the numerator:
\[ f(x) - g(x) = \frac{x - 1 - 1}{x} = \frac{x - 2}{x} \]

So, the simplified expression is:
\[ \boxed{\frac{x-2}{x}} \]

### 3. **\( f \cdot g \)**
\[ f(x) \cdot g(x) = \left(\frac{x-1}{x}\right) \cdot \left(\frac{1}{x}\right) \]

Multiply the numerators and the denominators:
\[ f(x) \cdot g(x) = \frac{(x-1) \cdot 1}{x \cdot x} = \frac{x-1}{x^2} \]

So, the simplified expression is:
\[ \boxed{\frac{x-1}{x^2}} \]

### 4. **\( \frac{f}{g} \)**
\[ \frac{f(x)}{g(x)} = \frac{\frac{x-1}{x}}{\frac{1}{x}} \]

To divide by a fraction, multiply by its reciprocal:
\[ \frac{f(x)}{g(x)} = \frac{x-1}{x} \cdot \frac{x}{1} \]

Simplifying:
\[ \frac{f(x)}{g(x)} = \frac{(x-1) \cdot x}{x \cdot 1} = x - 1 \]

So, the simplified expression is:
\[ \boxed{x-1} \]

### Summary of Simplified Expressions:
- \( f + g = 1 \)
- \( f - g = \frac{x-2}{x} \)
- \( f \cdot g = \frac{x-1}{x^2} \)
- \( \frac{f}{g} = x - 1 \)

Would you like more details or have any questions?

Here are five related questions you might find helpful:
1. How would you simplify \( f(x) + 2g(x) \)?
2. What is the result of \( f(x) \cdot (f(x) + g(x)) \)?
3. Can you find the simplified expression for \( (f(x))^2 - (g(x))^2 \)?
4. How do you simplify \( \frac{f(x) - g(x)}{f(x) + g(x)} \)?
5. What happens when you calculate \( \frac{f(x) \cdot g(x)}{f(x) + g(x)} \)?

**Tip**: When working with rational functions, always simplify the expression by combining like terms, multiplying, or factoring, and then cancel out common factors if possible.

Explore simplified expressions for rational functions f(x) = (x-1)/x and g(x) = 1/x, including f + g, f - g, f * g, and f / g. Learn how to combine fractions, simplify numerators and denominators, and understand the algebraic operations involved. Suitable for high school students studying algebra and rational functions.

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