https://file.aiquickdraw.com/mathbot/file/1731968819357dtawi8s8.jpg

Let's solve each of these problems by finding \( g(x) \) and \( h(x) \) such that \( f(x) = g(h(x)) \):

---

### **35(a) \( f(x) = \sqrt{x+2} \)**
We want \( f(x) = g(h(x)) \). Let:
- \( h(x) = x+2 \)
- \( g(x) = \sqrt{x} \)

Thus, \( g(h(x)) = \sqrt{h(x)} = \sqrt{x+2} \).

---

### **35(b) \( f(x) = |x^2 - 3x + 5| \)**
We want \( f(x) = g(h(x)) \). Let:
- \( h(x) = x^2 - 3x + 5 \)
- \( g(x) = |x| \)

Thus, \( g(h(x)) = |h(x)| = |x^2 - 3x + 5| \).

---

### **36(a) \( f(x) = x^2 + 1 \)**
We want \( f(x) = g(h(x)) \). Let:
- \( h(x) = x^2 \)
- \( g(x) = x + 1 \)

Thus, \( g(h(x)) = h(x) + 1 = x^2 + 1 \).

---

### **36(b) \( f(x) = \frac{1}{x-3} \)**
We want \( f(x) = g(h(x)) \). Let:
- \( h(x) = x-3 \)
- \( g(x) = \frac{1}{x} \)

Thus, \( g(h(x)) = \frac{1}{h(x)} = \frac{1}{x-3} \).

---

### **37(a) \( f(x) = \sin^2 x \)**
We want \( f(x) = g(h(x)) \). Let:
- \( h(x) = \sin(x) \)
- \( g(x) = x^2 \)

Thus, \( g(h(x)) = h(x)^2 = \sin^2 x \).

---

### **37(b) \( f(x) = \frac{3}{5+\cos x} \)**
We want \( f(x) = g(h(x)) \). Let:
- \( h(x) = 5 + \cos(x) \)
- \( g(x) = \frac{3}{x} \)

Thus, \( g(h(x)) = \frac{3}{h(x)} = \frac{3}{5+\cos x} \).

---

Would you like further clarifications or additional examples? Here are five related questions to consider:

1. Can you generalize the process of finding \( g(x) \) and \( h(x) \) for any given function \( f(x) \)?
2. What alternative choices for \( g(x) \) and \( h(x) \) could work for \( f(x) = \sqrt{x+2} \)?
3. How do we handle cases where \( f(x) \) includes composite trigonometric functions?
4. What happens if \( f(x) \) involves an absolute value with nested operations?
5. Can you verify these solutions by explicitly substituting \( h(x) \) into \( g(x) \)?

**Tip:** Always choose \( h(x) \) to simplify the innermost expression in \( f(x) \), then define \( g(x) \) accordingly.

Discover how to express functions as a composition of two functions with step-by-step solutions. Topics include square root functions, quadratic expressions, trigonometric identities, and absolute value functions. Ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation