91Ó°ÊÓ

The sum of functions involves adding two functions together. For instance, given two functions, \( f(m) = \frac{3}{m-4} \) and \( g(m) = \frac{-3m}{2m-5} \), their sum \( f(m) + g(m) \) is found by adding them:

First, find a common denominator for the fractions. Here, the common denominator is \((m-4)(2m-5)\).

Next, express each fraction with the common denominator:\
\( \frac{3(2m-5)}{(m-4)(2m-5)} + \frac{-3m(m-4)}{(m-4)(2m-5)} \).
Combine the numerators: \( \frac{3(2m-5) - 3m(m-4)}{(m-4)(2m-5)} \).
Simplify the numerator: \( 6m - 15 - 3m^2 + 12m \).
Combine like terms to get: \( \frac{-3m^2 + 18m - 15}{(m-4)(2m-5)} \).

Thus, the sum of the functions is \( f(m) + g(m) = \frac{-3m^2 + 18m - 15}{(m-4)(2m-5)} \).

To find the difference between two functions, subtract one function from the other. For the given functions \( f(m) = \frac{3}{m-4} \) and \( g(m) = \frac{-3m}{2m-5} \), their difference \( (f - g)(m) \) can be found by subtracting them:
Find a common denominator for the fractions, using \( (m-4)(2m-5) \), similarly as in the sum of functions.

Express each fraction with the common denominator:\
\( \frac{3(2m-5)}{(m-4)(2m-5)} - \frac{-3m(m-4)}{(m-4)(2m-5)} \).
Combine the numerators: \( \frac{3(2m-5) + 3m(m-4)}{(m-4)(2m-5)} \).
Simplify the numerator: \( 6m - 15 + 3m^2 - 12m \).
Combine like terms: \( \frac{3m^2 - 6m - 15}{(m-4)(2m-5)} \).

Thus, the difference of the functions is \( (f - g)(m) = \frac{3m^2 - 6m - 15}{(m-4)(2m-5)} \).

Finding the product of two functions involves multiplying them together. Given \( f(m) = \frac{3}{m-4} \) and \( g(m) = \frac{-3m}{2m-5} \), their product \( (f \cdot g)(m) \) is obtained as follows:
Multiply the numerators and the denominators of both functions: \( (f \cdot g)(m) = \frac{3}{m-4} \cdot \frac{-3m}{2m-5} \).
This simplifies to: \( \frac{3 \cdot -3m}{(m-4)(2m-5)} \).
Combine them to get: \( \frac{-9m}{(m-4)(2m-5)} \).

So, the product of the functions is \( (f \cdot g)(m) = \frac{-9m}{(m-4)(2m-5)} \).

The quotient of two functions is found by dividing one by the other. For the functions \( f(m) = \frac{3}{m-4} \) and \( g(m) = \frac{-3m}{2m-5} \), the quotient \( \left( \frac{f}{g} \right)(m) \) is determined as follows:

First, express the division as a multiplication by the reciprocal: \( \left( \frac{f}{g} \right)(m) = \frac{\frac{3}{m-4}}{\frac{-3m}{2m-5}} \).
This becomes: \( \frac{3}{m-4} \cdot \frac{2m-5}{-3m} \).
Multiply the numerators and denominators: \( \frac{3(2m-5)}{(m-4)(-3m)} \).
Simplify to: \( \frac{6m-15}{-3m(m-4)} \).

So, the quotient of the functions is \( \left( \frac{f}{g} \right)(m) = \frac{6m - 15}{-3m(m-4)} \).