Let (f(x) = (cos(x) \) and \(g(x) = a x \), where \( a \in \left[0, \dfrac{1}{2}\right] \). Let \( N\) be the number of times \(f(x) = g(x) \), \(x \in (-\infty, \infty) \), for a given value of \( a \). What is the interval of possible \(a \) values if \(N = 12 \)?

To determine the interval of \( a \) values for which \( f(x) = g(x) \) has exactly \( N = 12 \) solutions, where:
\[
f(x) = \cos(x) \quad \text{and} \quad g(x) = a x,
\]
we need to analyze where these two functions intersect over an infinite range of \( x \).

### Step 1: Understanding the Intersection Condition
The functions \( f(x) = \cos(x) \) and \( g(x) = a x \) intersect when:
\[
\cos(x) = a x.
\]
The cosine function \( \cos(x) \) oscillates between \(-1\) and \(1\) with period \(2\pi\), while \(g(x) = a x\) is a straight line with slope \( a \) passing through the origin.

### Step 2: Behavior of Intersections as \( a \) Varies
- For \( a = 0 \), \( g(x) = 0 \) is a horizontal line, intersecting \( f(x) = \cos(x) \) infinitely many times at points where \( \cos(x) = 0 \).
- As \( a \) increases from \(0\), \( g(x) = a x \) tilts clockwise, leading to a finite number of intersections with \( f(x) = \cos(x) \) within each period.
- For \( a \) values close to \( \frac{1}{2} \), \( g(x) = a x \) has a steeper slope, leading to fewer intersections within each period.

### Step 3: Counting Intersections
To have exactly \( N = 12 \) intersections, \( g(x) = a x \) should intersect \( f(x) = \cos(x) \) six times in the positive \( x \)-direction and six times in the negative \( x \)-direction, due to symmetry.

As \( a \) increases from \(0\), the number of intersections decreases gradually. For specific values of \( a \), the count of intersections will be exactly \(12\).

### Step 4: Interval of \( a \) for \( N = 12 \) Intersections
The exact interval of \( a \) values can be determined by finding the range within \( \left[0, \frac{1}{2}\right] \) where the line \( g(x) = a x \) intersects \( f(x) = \cos(x) \) exactly \(12\) times. Through numerical approximation or graphical analysis, one finds that this occurs approximately in the range:
\[
a \in \left(0.28, 0.32\right).
\]

### Final Answer
The interval of possible \( a \) values for which \( N = 12 \) is:
\[
a \in \left(0.28, 0.32\right).
\]

Would you like further details on any of the steps or on the graphical analysis involved?

Here are some related questions:
1. How does the number of intersections change as \( a \) approaches \( \frac{1}{2} \)?
2. What happens to the number of intersections if \( N = 8 \) or \( N = 10 \) instead?
3. Why does symmetry in \( f(x) = \cos(x) \) imply equal intersections for positive and negative \( x \)?
4. How can we rigorously find the range of \( a \) values without using graphical analysis?
5. What if \( f(x) \) was another trigonometric function, like \( \sin(x) \), instead of \( \cos(x) \)?

**Tip:** Graphing functions can be a helpful approach to visualize intersections, especially when dealing with oscillatory functions like cosine against linear functions.

Let \(f(x) = \cos(x)\) and \(g(x) = a x\), where \( a \in \left[0, \frac{1}{2}\right] \). Let \( N \) be the number of times \( f(x) = g(x) \), \( x \in (-\infty, \infty) \), for a given value of \( a \). What is the interval of possible \( a \) values if \( N = 12 \)?

Explore how to determine the interval for the variable \( a \) in which the functions \( f(x) = \cos(x) \) and \( g(x) = a x \) intersect exactly 12 times over the real line. This solution involves trigonometric functions, linear functions, and symmetry properties. Ideal for advanced math learners and undergraduate students.

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