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.