For questions 5 through 10 , consider $$ m(x)=\frac{2}{x}, t(x)=2 x^{2}+3 x, w(x)=2 x+5 $$. 5. Determine $$ (m \circ t)(2) $$. 6. Determine $$ (t \circ w)(x) $$. 7. Determine the domain of $$ (t \circ w)(x) $$. Determine $$ (t \circ m)(x) $$. Determine the domain of $$ (t \circ m)(x) $$.

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Let's address each of the questions step by step using the given functions:

- $$ m(x) = \frac{2}{x} $$
- $$ t(x) = 2x^2 + 3x $$
- $$ w(x) = 2x + 5 $$

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### **5. Determine $$ (m \circ t)(2) $$**

**Definition:**
$$ (m \circ t)(x) = m(t(x)) $$

**Calculation:**
1. **Find $$ t(2) $$:**
   $$
   t(2) = 2(2)^2 + 3(2) = 2(4) + 6 = 8 + 6 = 14
   $$
2. **Find $$ m(t(2)) = m(14) $$:**
   $$
   m(14) = \frac{2}{14} = \frac{1}{7}
   $$

**Answer:**
$$
(m \circ t)(2) = \frac{1}{7}
$$

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### **6. Determine $$ (t \circ w)(x) $$**

**Definition:**
$$ (t \circ w)(x) = t(w(x)) $$

**Calculation:**
1. **Express $$ t(w(x)) $$:**
   $$
   t(w(x)) = 2(w(x))^2 + 3w(x) = 2(2x + 5)^2 + 3(2x + 5)
   $$
2. **Expand and simplify:**
   $$
   (2x + 5)^2 = 4x^2 + 20x + 25
   $$
   $$
   t(w(x)) = 2(4x^2 + 20x + 25) + 3(2x + 5) = 8x^2 + 40x + 50 + 6x + 15 = 8x^2 + 46x + 65
   $$

**Answer:**
$$
(t \circ w)(x) = 8x^2 + 46x + 65
$$

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### **7. Determine the Domain of $$ (t \circ w)(x) $$**

**Analysis:**
- **Function $$ w(x) = 2x + 5 $$:** Defined for all real numbers.
- **Function $$ t(x) = 2x^2 + 3x $$:** Also defined for all real numbers since it's a polynomial.

**Conclusion:**
Since both $$ w(x) $$ and $$ t(x) $$ are defined for all real numbers, their composition $$ t(w(x)) $$ is also defined for all real numbers.

**Answer:**
$$
\text{Domain of } (t \circ w)(x) = \mathbb{R} \quad \left(-\infty,\, \infty\right)
$$

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### **8. Determine $$ (t \circ m)(x) $$**

**Definition:**
$$ (t \circ m)(x) = t(m(x)) $$

**Calculation:**
1. **Express $$ m(x) $$:**
   $$
   m(x) = \frac{2}{x}
   $$
2. **Substitute into $$ t(x) $$:**
   $$
   t\left(\frac{2}{x}\right) = 2\left(\frac{2}{x}\right)^2 + 3\left(\frac{2}{x}\right) = 2\left(\frac{4}{x^2}\right) + \frac{6}{x} = \frac{8}{x^2} + \frac{6}{x}
   $$

**Simplified Form:**
$$
(t \circ m)(x) = \frac{8}{x^2} + \frac{6}{x} \quad \text{or} \quad \frac{8 + 6x}{x^2}
$$

**Answer:**
$$
(t \circ m)(x) = \frac{8}{x^2} + \frac{6}{x}
$$

---

### **9. Determine the Domain of $$ (t \circ m)(x) $$**

**Analysis:**
- **Function $$ m(x) = \frac{2}{x} $$:** Defined for all $$ x $$ except $$ x = 0 $$.
- **Function $$ t(x) = 2x^2 + 3x $$:** Defined for all real numbers.

**Composition $$ t(m(x)) $$:**
- The inner function $$ m(x) $$ must be defined, which requires $$ x \neq 0 $$.
- Since $$ t(x) $$ is defined for all real numbers, no additional restrictions are imposed.

**Conclusion:**
The domain of $$ (t \circ m)(x) $$ consists of all real numbers except $$ x = 0 $$.

**Answer:**
$$
\text{Domain of } (t \circ m)(x) = \mathbb{R} \setminus \{0\} \quad \left(-\infty,\, 0\right) \cup \left(0,\, \infty\right)
$$

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Feel free to ask if you need further clarifications or additional assistance!
$$ (m \circ t)(2) = \frac{1}{7} $$ $$ (t \circ w)(x) = 8x^2 + 46x + 65 $$ $$ \text{Domain of } (t \circ w)(x) = \mathbb{R} $$ $$ (t \circ m)(x) = \frac{8}{x^2} + \frac{6}{x} $$ $$ \text{Domain of } (t \circ m)(x) = \mathbb{R} \setminus \{0\} $$

For questions 5 through 10 , consider \( m(x)=\frac{2}{x}, t(x)=2 x^{2}+3 x, w(x)=2 x+5 \). 5. Determine \( (m \circ t)(2) \). 6. Determine \( (t \circ w)(x) \). 7. Determine the domain of \( (t \circ w)(x) \). Determine \( (t \circ m)(x) \). Determine the domain of \( (t \circ m)(x) \).

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