Express

​f(x)plus+​g(x)

as a rational function. Carry out all multiplications.

​f(x)equals=StartFraction 2 Over x minus 5 EndFraction2x−5

​g(x)equals=StartFraction 3 Over x plus 4 EndFraction3x+4

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Part 1

​f(x) +​ g(x) =

To express \( f(x) + g(x) \) as a rational function, we need to add the two given rational functions:

\[
f(x) = \frac{2}{x - 5}, \quad g(x) = \frac{3}{x + 4}
\]

### Step 1: Find a common denominator.
The denominators are \( x - 5 \) and \( x + 4 \), so the common denominator will be the product of these two:

\[
\text{Common denominator} = (x - 5)(x + 4)
\]

### Step 2: Rewrite each fraction with the common denominator.

For \( f(x) \):

\[
f(x) = \frac{2}{x - 5} = \frac{2(x + 4)}{(x - 5)(x + 4)}
\]

For \( g(x) \):

\[
g(x) = \frac{3}{x + 4} = \frac{3(x - 5)}{(x - 5)(x + 4)}
\]

### Step 3: Add the two fractions.

\[
f(x) + g(x) = \frac{2(x + 4)}{(x - 5)(x + 4)} + \frac{3(x - 5)}{(x - 5)(x + 4)}
\]

Since they now have a common denominator, we can add the numerators:

\[
f(x) + g(x) = \frac{2(x + 4) + 3(x - 5)}{(x - 5)(x + 4)}
\]

### Step 4: Simplify the numerator.

Distribute the constants in the numerator:

\[
2(x + 4) = 2x + 8, \quad 3(x - 5) = 3x - 15
\]

Now combine like terms:

\[
2x + 8 + 3x - 15 = 5x - 7
\]

### Step 5: Write the final expression.

\[
f(x) + g(x) = \frac{5x - 7}{(x - 5)(x + 4)}
\]

Thus, the rational function is:

\[
f(x) + g(x) = \frac{5x - 7}{(x - 5)(x + 4)}
\]

Would you like more details or have any questions?

Here are some related questions to practice:
1. How do you subtract two rational functions?
2. What happens when the denominators of two rational functions are the same?
3. How do you multiply two rational functions?
4. What is the process for simplifying a complex rational function?
5. How do you find the domain of a rational function?

**Tip**: When adding or subtracting rational functions, always look for a common denominator first.

Express f(x) + g(x) as a rational function. Carry out all multiplications.
f(x) = 2/(x - 5), g(x) = 3/(x + 4)

Learn how to add two rational functions, f(x) = 2/(x - 5) and g(x) = 3/(x + 4), by finding a common denominator and simplifying the expression. This step-by-step solution demonstrates key algebraic concepts, such as the addition of fractions and rational functions. Suitable for students in Grades 9-11.

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