given the functions f(x)=cosx and g(x)=1/3, determine the range of the combined function f(x)/g(x)

Given the functions \( f(x) = \cos(x) \) and \( g(x) = \frac{1}{3} \), we need to determine the range of the combined function \( h(x) = \frac{f(x)}{g(x)} \).

### Step-by-Step Solution

1. **Identify the functions**:
   - \( f(x) = \cos(x) \)
   - \( g(x) = \frac{1}{3} \)

2. **Determine the range of \( f(x) \)**:
   - The function \( f(x) = \cos(x) \) oscillates between \(-1\) and \(1\).
   - Therefore, the range of \( f(x) \) is \( [-1, 1] \).

3. **Determine the value of \( g(x) \)**:
   - The function \( g(x) = \frac{1}{3} \) is a constant, so it always equals \( \frac{1}{3} \).

4. **Form the combined function \( h(x) = \frac{f(x)}{g(x)} \)**:
   - Substituting the functions into the combined function gives:
     \[
     h(x) = \frac{\cos(x)}{\frac{1}{3}} = 3 \cos(x)
     \]

5. **Determine the range of \( h(x) \)**:
   - Since \( \cos(x) \) ranges from \(-1\) to \(1\), multiplying by \(3\) scales this range by a factor of 3.
   - The new range of \( h(x) \) is:
     \[
     h(x) = 3 \cos(x) \text{ ranges from } 3(-1) \text{ to } 3(1)
     \]
   - Therefore, the range of \( h(x) = 3 \cos(x) \) is \( [-3, 3] \).

### Final Answer:
The range of the combined function \( h(x) = \frac{f(x)}{g(x)} = \frac{\cos(x)}{\frac{1}{3}} = 3 \cos(x) \) is \( [-3, 3] \).

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Would you like further details on any part of this solution? Here are some related questions you might find interesting:

1. How does the range of \( f(x) \) change if \( g(x) \) is a different constant, say \( \frac{1}{2} \)?
2. What happens to the range if \( g(x) \) is a function instead of a constant?
3. How do you determine the range of a function like \( h(x) = \cos(x) \times \frac{1}{3} \)?
4. What would the range be if \( g(x) = 2 \) instead of \( \frac{1}{3} \)?
5. Can you determine the range of a combined function like \( h(x) = f(x) + g(x) \) when \( g(x) \) is a constant?

**Tip:** When dividing by a constant, the range of the original function is simply scaled by the reciprocal of that constant.

Explore how to determine the range of the combined function f(x)/g(x) where f(x) = cos(x) and g(x) = 1/3. This problem involves understanding trigonometric functions, division of functions, and range determination in mathematics. Suitable for high school students.

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