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When discussing the domain of a function, we refer to all the possible input values (or x-values) that the function can accept. For the reciprocal function \( f(x) = \frac{1}{(x-4)^2} \), determining the domain involves identifying the x-values that do not cause the function to become undefined.

In this case, the denominator \((x-4)^2\) cannot be zero, because division by zero is undefined in mathematics. Thus, setting the denominator equal to zero, we solve \(x-4 = 0\) to find that \(x = 4\) is the point where the function is undefined. Consequently, the domain of this function includes all real numbers except \(x = 4\).

This can be expressed in interval notation as

• \((-∞, 4)\)
• \((4, ∞)\)

indicating that the function accepts every real number as input apart from 4.

The range of a function consists of all possible output values (or y-values) that the function can produce after processing all permitted x-values from its domain. For the function \( f(x) = \frac{1}{(x-4)^2} \), analyzing the range involves understanding how the function behaves given the constraints of its denominator.

Because the denominator \((x-4)^2\) is a squared term, it is always positive for all numbers except x = 4, where the function is undefined. The numerator, being a constant value of 1, ensures that the entire function remains positive for all legitimate inputs. Moreover, since \((x-4)^2\) can become arbitrarily large as \(x\) moves away from 4 in either direction, the value of the entire fraction becomes very small, approaching but never reaching 0.

Therefore, the range of \( f(x) \) includes all positive real numbers and can be expressed in interval notation as

• \((0, ∞)\)

indicating that the function outputs any positive number but never zero.

An even function is characterized by its symmetry across the y-axis. This means that for an even function \( f \), the relationship \( f(-x) = f(x) \) holds true for all x in the function's domain. In the case of our function \( f(x) = \frac{1}{(x-4)^2} \), it is important to consider how the function reacts to positive and negative inputs.

Given the squared term \((x-4)^2\), any negative deviation from \(x = 4\) yields the same result as a positive deviation, because squaring negates any negative sign. Thus, \( f(x) \) is identical whether x is substituted by \(+a\) or \(-a\) (where a is the distance from 4). This kind of behavior is a hallmark of even functions as they naturally `mirror` on either side of the y-axis.

Understanding this symmetry helps in quickly identifying and graphing even functions while also providing insights into their range and other properties. This trait makes even functions predictable in how they handle positive and negative inputs.