If f(x) = 1/(3x + 4) and g(x) = 1/(x - 2), what is f + g?

Steps to Solve

1. Find a Common Denominator

To add the two functions $f(x)$ and $g(x)$, we need a common denominator. The functions are:

$$ f(x) = \frac{1}{3x + 4} $$

$$ g(x) = \frac{1}{x - 2} $$

The common denominator will be the product of both denominators:

$$ (3x + 4)(x - 2) $$

2. Rewrite Each Function

Now we will rewrite each function with the common denominator:

For $f(x)$, multiply the numerator and denominator by $(x - 2)$:

$$ f(x) = \frac{1 \cdot (x - 2)}{(3x + 4)(x - 2)} = \frac{x - 2}{(3x + 4)(x - 2)} $$

For $g(x)$, multiply the numerator and denominator by $(3x + 4)$:

$$ g(x) = \frac{1 \cdot (3x + 4)}{(x - 2)(3x + 4)} = \frac{3x + 4}{(3x + 4)(x - 2)} $$

3. Add the Two Functions

Now we can sum the two rewritten functions:

$$ f(x) + g(x) = \frac{x - 2}{(3x + 4)(x - 2)} + \frac{3x + 4}{(3x + 4)(x - 2)} $$

Combine the numerators:

$$ f(x) + g(x) = \frac{(x - 2) + (3x + 4)}{(3x + 4)(x - 2)} $$

4. Simplify the Numerator

Combine like terms in the numerator:

$$ (x - 2) + (3x + 4) = 4x + 2 $$

So now we have:

$$ f(x) + g(x) = \frac{4x + 2}{(3x + 4)(x - 2)} $$