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The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the given function, \(y = \ln(-x) + e\), we must consider the nature of the logarithmic component. Since \(\ln(x)\) is defined only for positive numbers, \(-x\) must be greater than zero, implying \(x < 0\). Therefore, the domain is all negative real numbers.
It's important to note that the function is not defined at \(x = 0\), further emphasizing the restriction. Because of this, our domain is restricted to negative real numbers, expressed as \(x < 0\).
The range of a function is the set of all possible output values (y-values). Typically, the natural logarithm function, \(\ln(x)\), spans all real numbers from negative to positive infinity. Although our function has been vertically shifted by \(e\) units, this does not alter the range's infinite behavior. Thus, the range of the function \(y = \ln(-x) + e\) remains \((−\infty, \infty)\).

Function transformations involve altering a function's graph through shifting, stretching, compressing, or reflecting its original form. In the function \(y = \ln(-x) + e\), we can observe two key transformations:

1. **Reflection across the y-axis:**

• The natural logarithm function \(\ln(x)\) is reflected across the y-axis because of the negative sign in the argument \(-x\).

This flip changes the orientation of the graph, causing it to open towards the negative x-axis.

2. **Vertical shift:**

• The entire graph is shifted upward by \(e\) units due to the \(+ e\) outside the logarithmic function.

This changes the baseline height of all the y-values, without altering the domain or the range significantly. As a result, the entire graph of \(\ln(-x)\) like a mirror image is moved "up" the y-axis by approximately 2.718 units.

Asymptotes are lines that a graph gets infinitely close to, but never actually touches. For the function \(y = \ln(-x) + e\), we focus on the vertical asymptote:

The function \(\ln(-x)\) has a critical point where \(-x = 0\), meaning at \(x = 0\), the function is undefined. This results in a vertical asymptote at the y-axis (line \(x = 0\)).

This vertical asymptote is significant because it represents an invisible barrier on the graph. As the function approaches \(x = 0\) from the left-hand side (negative x-values), the graph will climb steeply but never intersect the line. This behavior helps to visually represent and remind us why the domain is \(x < 0\).

Intercepts are the points where a graph crosses the x-axis or y-axis. Identifying these values for \(y = \ln(-x) + e\) can reveal a lot about the function's behavior.

**Y-intercept:**
To find a y-intercept, we set \(x = 0\) and solve for \(y\). However, since \(\ln(-x)\) becomes \(\ln(0)\) which is undefined, there is no y-intercept for this function.

**X-intercept:**
Setting \(y = 0\) allows us to find the x-intercept. We solve:
\[\ln(-x) + e = 0\]
Subtract \(e\) from both sides to get \(\ln(-x) = -e\).
Exponentiating both sides, we find that \(-x = e^{-e}\) or \(x = -e^{-e}\).
Thus, the graph crosses the x-axis at the point \((-e^{-e}, 0)\).

This single x-intercept gives us a concrete point where the graph of the function intersects the horizontal axis.