Find the (h+t)(x) which is h(x)+t(x), the given is h(x)=(x+7/2-x) and t(x)=(x-2/x+3)?step by step.

To find the function \((h + t)(x)\), we need to add the functions \(h(x)\) and \(t(x)\) together. Let's break it down step by step.

### Step 1: Write down the given functions
We are given:
\[
h(x) = \left(x + \frac{7}{2} - x\right)
\]
\[
t(x) = \left(\frac{x - 2}{x + 3}\right)
\]

### Step 2: Simplify the function \(h(x)\)
Simplify the expression for \(h(x)\) by combining like terms:
\[
h(x) = x + \frac{7}{2} - x
\]
The \(x\) and \(-x\) cancel each other out:
\[
h(x) = \frac{7}{2}
\]

So, \(h(x)\) is a constant function:
\[
h(x) = \frac{7}{2}
\]

### Step 3: Add the functions \(h(x)\) and \(t(x)\)
Now, let's add \(h(x)\) and \(t(x)\) to find \((h + t)(x)\):
\[
(h + t)(x) = h(x) + t(x) = \frac{7}{2} + \frac{x - 2}{x + 3}
\]

### Step 4: Combine the expressions
To combine these into a single expression, we'll write it as:
\[
(h + t)(x) = \frac{7}{2} + \frac{x - 2}{x + 3}
\]

To combine these into a single fraction, we need a common denominator. The common denominator is \(2(x + 3)\):
\[
(h + t)(x) = \frac{7(x + 3)}{2(x + 3)} + \frac{2(x - 2)}{2(x + 3)}
\]

### Step 5: Simplify the expression
Distribute and simplify the numerators:
\[
h(x) + t(x) = \frac{7(x + 3) + 2(x - 2)}{2(x + 3)}
\]
\[
= \frac{7x + 21 + 2x - 4}{2(x + 3)}
\]
\[
= \frac{9x + 17}{2(x + 3)}
\]

### Final Answer:
\[
(h + t)(x) = \frac{9x + 17}{2(x + 3)}
\]

This is the simplified form of the function \((h + t)(x)\).

Would you like further details on any of these steps or any additional explanations?

---

Here are 8 related questions to explore further:

1. How would you subtract \(t(x)\) from \(h(x)\)?
2. What is the domain of the function \((h + t)(x)\)?
3. How would you find the roots of the function \((h + t)(x)\)?
4. Can \((h + t)(x)\) be simplified further?
5. How does the behavior of \((h + t)(x)\) change as \(x\) approaches infinity?
6. How would you determine the inverse of \((h + t)(x)\)?
7. What are the vertical and horizontal asymptotes of \((h + t)(x)\)?
8. How would you graph the function \((h + t)(x)\)?

**Tip:** When adding fractions, always find a common denominator to combine them into a single expression.

Learn how to find the sum (h + t)(x) where h(x) = (x + 7/2 - x) and t(x) = (x - 2) / (x + 3). Follow a detailed step-by-step solution to combine the functions and simplify the result. Suitable for high school students and anyone studying functions and algebraic operations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation