can you help me evaluate the function $$ \left(f+h\right)\left(-4\right)\text{for the given value of }x\text{ }f\left(x\right)=3x\text{   }h\left(x\right)=\frac{1}{x+7} $$

To evaluate $(f + h)(-4)$, we need to first define what this function represents.

We know that $ (f + h)(x) = f(x) + h(x) $. So, in this case:

\[
(f + h)(-4) = f(-4) + h(-4)
\]

Now, let's calculate $ f(-4) $ and $ h(-4) $ individually.

### Step 1: Calculate $ f(-4) $
Given that $ f(x) = 3x $, we have:

\[
f(-4) = 3(-4) = -12
\]

### Step 2: Calculate $ h(-4) $
Given that $ h(x) = \frac{1}{x + 7} $, we have:

\[
h(-4) = \frac{1}{-4 + 7} = \frac{1}{3}
\]

### Step 3: Add the results
Now we add $ f(-4) $ and $ h(-4) $:

\[
(f + h)(-4) = -12 + \frac{1}{3}
\]

To simplify, we can write this as:

\[
(f + h)(-4) = -12 + \frac{1}{3} = \frac{-36}{3} + \frac{1}{3} = \frac{-35}{3}
\]

Thus, the value of $ (f + h)(-4) $ is:

\[
(f + h)(-4) = \frac{-35}{3}
\]

Would you like more details on any of the steps?

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Here are five related questions you could explore:

1. What is $ (f + h)(0) $?
2. How do you calculate $ (f \cdot h)(-4) $?
3. What happens to $ h(x) $ when $ x = -7 $?
4. Can you find the derivative of $ f(x) + h(x) $?
5. What is the domain of $ h(x) $?

**Tip:** Always be cautious of division by zero when working with rational functions like $ h(x) = \frac{1}{x+7} $.

Can you help me evaluate the function (f+h)(-4) for the given value of x, where f(x)=3x and h(x)=\frac{1}{x+7}?

Learn how to evaluate the combined function (f+h)(-4) for f(x)=3x and h(x)=1/(x+7). This algebra problem involves function operations and rational expressions, suitable for students in Grades 9-12.

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