Question

Use a graphing calculator to graph the function. Determine the domain, range, and asymptote of the function. \(y=\ln (-x)\)

Short answer

The graph of the function \(y=\ln (-x)\) is a reflection of the graph of \(y=\ln (x)\) in the y-axis. The domain of the function is \((-\infty, 0)\), the range is \((-\infty ,+\infty)\), and there is a vertical asymptote at x=0.

Step-by-step solution

Graphing the function

Input the function \(y=\ln (-x)\) into a graphing calculator and generate its graph. If not available, use an online graphing calculator tool.

Finding the domain

The domain of a function is the set of x-values for which the function is defined. The natural logarithm is only defined for x-values greater than 0. Since there is an impact of negative sign, the domain is all x-values that are less than 0. Therefore, the domain for the function \(y=\ln (-x)\) is \((-\infty, 0)\).

Finding the range

The range of a function is the set of y-values that the function can take. Since the natural logarithm can output any real number, the range of \(y=\ln (-x)\) is indeed \( (-\infty ,+\infty)\).

Determining the asymptote

Vertical asymptotes occur at the x-values where the function is undefined. For the function \(y=\ln (-x)\), there is a vertical asymptote at x=0, as approaching 0 from the left (i.e., with negative x-values) results in the function values approaching negative infinity, hence x=0 is the vertical asymptote.

Key concepts

Domain of a Function

When graphing logarithmic functions, understanding the domain of a function is crucial. The domain encompasses all the real numbers that can be input into the function, allowing it to work properly without producing errors or undefined values. In the case of the natural logarithm function, which is written as \(y=\ln(-x)\), the domain is a little trickier because you cannot take the logarithm of a negative number or zero.

To find the domain of \(y=\ln(-x)\), we must consider what values of \(x\) will make \(\ln(-x)\) defined. This occurs when \(\ln\) has a positive input, meaning that \(\-x > 0\). Solving for \(x\), we get \(x < 0\), which means that the domain of \(y=\ln(-x)\) is \( (-\infty, 0) \). Put simply, we can input any negative number into our function, but nothing zero or positive.

Range of a Function

Equally important as the domain, the range of a function defines all the possible outputs or \(y\)-values a function can have. For logarithmic functions like \(y=\ln(-x)\), the range is particularly straightforward: it's all real numbers. No matter what negative value you input for \(x\), the output can be any real number from negative to positive infinity.

This is because the natural logarithm function, at its essence, measures the time it takes to grow to a certain level and can indeed take an infinite amount of time (approaching infinity) or grow instantly (approaching negative infinity). Therefore, the range for our function \(y=\ln(-x)\) is \( (-\infty, +\infty)\). No matter how large or small a negative value you input (within the function's domain), you can receive an output as high or low as you can imagine.

Vertical Asymptote

Definition and Identification

A vertical asymptote is essentially a boundary that a function's graph approaches but never reaches or crosses. For the function \(y=\ln(-x)\), we find a vertical asymptote at \(x=0\). This is because as \(x\) nears zero from the left (negative side), the values of \(y\) shoot off towards negative infinity. The graph gets closer and closer to the line \(x=0\) but doesn't touch it, creating a sort of 'forbidden line' that dictates where the graph can go.

Understanding and identifying these asymptotes is important as they are critical features of the graph's behavior and shape, adding to a full understanding of the function's properties.

Natural Logarithm

The natural logarithm can seem daunting, but it's a fundamental concept in mathematics. Denoted as \(\ln(x)\), it is the logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. The function \(\ln(x)\) is the inverse of the exponential function \(e^x\).

For the function in our exercise \(y=\ln(-x)\), the natural logarithm translates into 'the power to which we must raise \(e\) to get \(\-x\) as the result'. It's important to recognize that the concept of the natural logarithm seamlessly integrates with not just other forms of logarithms but also with exponential functions, compounding interest in finance, and natural growth or decay processes in science.