Math

Question Express f(x)+g(x)f(x)+g(x) as a simplified rational function. Given f(x)=6x7,g(x)=5x+6f(x)=\frac{6}{x-7}, g(x)=\frac{5}{x+6}, find f(x)+g(x)f(x)+g(x).

Studdy Solution

STEP 1

Assumptions
1. The function f(x)f(x) is given as 6x7\frac{6}{x-7}.
2. The function g(x)g(x) is given as 5x+6\frac{5}{x+6}.
3. To find f(x)+g(x)f(x) + g(x), we need to add the two functions together.
4. The sum of the functions should be expressed as a single rational function.

STEP 2

To add two rational functions, we need a common denominator. In this case, the common denominator will be the product of the two denominators (x7)(x+6)(x-7)(x+6).

STEP 3

Rewrite f(x)f(x) with the common denominator.
f(x)=6x7=6(x+6)(x7)(x+6)f(x) = \frac{6}{x-7} = \frac{6(x+6)}{(x-7)(x+6)}

STEP 4

Rewrite g(x)g(x) with the common denominator.
g(x)=5x+6=5(x7)(x7)(x+6)g(x) = \frac{5}{x+6} = \frac{5(x-7)}{(x-7)(x+6)}

STEP 5

Now that both functions have the same denominator, we can add the numerators together.
f(x)+g(x)=6(x+6)(x7)(x+6)+5(x7)(x7)(x+6)f(x) + g(x) = \frac{6(x+6)}{(x-7)(x+6)} + \frac{5(x-7)}{(x-7)(x+6)}

STEP 6

Combine the numerators over the common denominator.
f(x)+g(x)=6(x+6)+5(x7)(x7)(x+6)f(x) + g(x) = \frac{6(x+6) + 5(x-7)}{(x-7)(x+6)}

STEP 7

Distribute the numbers across the terms in the numerators.
f(x)+g(x)=6x+36+5x35(x7)(x+6)f(x) + g(x) = \frac{6x + 36 + 5x - 35}{(x-7)(x+6)}

STEP 8

Combine like terms in the numerator.
f(x)+g(x)=11x+1(x7)(x+6)f(x) + g(x) = \frac{11x + 1}{(x-7)(x+6)}

STEP 9

The expression 11x+1(x7)(x+6)\frac{11x + 1}{(x-7)(x+6)} is already simplified, so this is our final answer.
f(x)+g(x)=11x+1(x7)(x+6)f(x) + g(x) = \frac{11x + 1}{(x-7)(x+6)}

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