if f(g(x))= ((|x|)/(2((x^2)-1))) and g(x)= 4(x^2) +1 then what is the domain of f(x)

We are given the functions \( f(g(x)) = \frac{|x|}{2(x^2 - 1)} \) and \( g(x) = 4x^2 + 1 \), and the task is to find the domain of \( f(x) \).

### Step 1: Analyze \( g(x) \)

The function \( g(x) = 4x^2 + 1 \) is a quadratic function. Since \( 4x^2 \) is non-negative for all real \( x \), and adding 1 makes the function strictly positive, \( g(x) \) is always positive. Therefore, the domain of \( g(x) \) is all real numbers \( \mathbb{R} \), as there are no restrictions.

### Step 2: Analyze \( f(g(x)) \)

The function \( f(g(x)) = \frac{|x|}{2(x^2 - 1)} \) depends on two factors:
1. The denominator \( 2(x^2 - 1) \) cannot be zero because division by zero is undefined.
2. The absolute value \( |x| \) in the numerator does not impose any restrictions on the domain.

For \( 2(x^2 - 1) = 0 \), the denominator becomes zero when:
\[
x^2 - 1 = 0 \quad \Rightarrow \quad x^2 = 1 \quad \Rightarrow \quad x = \pm 1
\]
Thus, \( f(g(x)) \) is undefined at \( x = 1 \) and \( x = -1 \).

### Step 3: Determine the domain of \( f(x) \)

The function \( f(g(x)) \) is defined for all real numbers except \( x = 1 \) and \( x = -1 \), as these values make the denominator zero.

Therefore, the domain of \( f(x) \) is:
\[
\boxed{(-\infty, -1) \cup (-1, 1) \cup (1, \infty)}
\]

Would you like more details, or do you have any further questions?

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Here are 5 related questions to deepen understanding:

1. What is the range of the function \( g(x) = 4x^2 + 1 \)?
2. How does the absolute value in \( f(g(x)) \) affect its behavior for negative values of \( x \)?
3. How would the domain change if \( g(x) = x^2 - 4 \) instead of \( 4x^2 + 1 \)?
4. Can you describe how to find the inverse of \( f(x) \) if it exists?
5. How would the domain of \( f(x) \) be affected if the denominator included a square root?

**Tip:** When working with rational functions, always check for points where the denominator becomes zero, as these are restrictions on the domain.

Explore how to find the domain of the composite function f(g(x)) = |x| / (2(x^2 - 1)) where g(x) = 4x^2 + 1. This detailed step-by-step guide covers the key concepts of domain restrictions, absolute values, and rational functions, suitable for high school students in Grades 10-12.

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