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 $$ (t \circ m)(x) $$.

Question

UpStudy AI · Accepted Answer

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

---

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

---

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

---

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

---

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 .
5. Determine .
6. Determine .
7. Determine the domain of .

Determine .

Determine the domain of .

Upstudy AI Solution

Answer

Solution

5. Determine

6. Determine

7. Determine the Domain of

8. Determine

9. Determine the Domain of

Bonus Knowledge

5. Determine .

6. Determine .

7. Determine the domain of .

8. Determine .

9. Determine the domain of .

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