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The domain of a function refers to all the possible input values (or x-values) that you can plug into a function without causing any problems like division by zero or taking the square root of a negative number. For the function \( y = 1 + \frac{1}{(x+2)^2} \), the denominator \((x+2)^2\) cannot be zero, as division by zero is undefined in mathematics. The expression \((x+2)^2\) equals zero when \(x = -2\). Therefore, this particular x-value is excluded from the domain of the function. For this function, the domain includes all real numbers except \(x = -2\).

To summarize:

• The domain is all real numbers except \(x = -2\).

Identifying the domain is crucial for understanding where the function exists and behaves smoothly.

Asymptotes are lines that a graph approaches but never actually touches. They help you understand the behavior of functions as they move toward infinity. For this function, there are two types of asymptotes: vertical and horizontal.

**Vertical Asymptotes**
The vertical asymptote of this function is at \(x = -2\), determined by where the function is undefined. As \(x\) gets closer to \(-2\), without actually touching it, the values of \(y\) shoot up towards infinity. This behavior is typical near vertical asymptotes, where the graph of a function experiences a sort of "break."

**Horizontal Asymptotes**
A horizontal asymptote looks at the behavior of the function as \(x\) goes towards positive or negative infinity. For this function, the horizontal asymptote is \(y = 1\). As \(x\) moves towards very large positive or negative numbers, \(\frac{1}{(x+2)^2}\) tends toward zero, making \(y\) approach 1. So, the graph will get closer and closer to this y-value but never actually reach it.

Intercepts are the points where the graph of a function crosses the x-axis or y-axis. These points give us a snapshot of key intersections in the graph, which are crucial for sketching it.

**Y-intercept**
To find the y-intercept, we set \(x = 0\) in the function, giving us \(y = 1 + \frac{1}{4} = 1.25\). Thus, the graph crosses the y-axis at the point \((0, 1.25)\). This point helps in positioning the overall graph on the coordinate plane.

**X-intercept**
An x-intercept occurs when \(y = 0\). Solving \(1 + \frac{1}{(x+2)^2} = 0\), we find there is no solution, meaning the graph never crosses the x-axis. The absence of an x-intercept is important because it confirms the graph doesn't dip below the x-axis.

Critical points in a function's graph are where you might experience abrupt changes or an interesting behavior, such as the graph shooting toward infinity or settling into a flat line.

**Near the Vertical Asymptote**
The behavior of the function \(y = 1 + \frac{1}{(x+2)^2}\) around the vertical asymptote \(x = -2\) is telling. As \(x\) approaches \(-2\) from the left \((x \to -2^-)\), the values of \(y\) increase sharply toward positive infinity. Similarly, from the right \((x \to -2^+)\), \(y\) also rises rapidly towards positive infinity. This simultaneous rise on both sides reflects the distinct nature of a vertical asymptote.

**Approaching the Horizontal Asymptote**
As \(x\) moves away in either direction from the asymptote \(x = -2\), the graph stabilizes and approaches the horizontal asymptote \(y=1\). Whether \(x\) is much larger or smaller than \(-2\), \(y\) gently decreases and approaches \(y=1\), highlighting the role of horizontal asymptotes in anchoring the long-term behavior of a function's graph.