Problem 69 - College Algebra

Question

The figure shows the graph of \(f(x)=\ln x\). Use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. (GRAPH CANNOT COPY). $$ g(x)=2 \ln x $$

Step-by-Step Solution

Answer

The graph of \(g(x) = 2 \ln x\) is the graph of the original function \(\ln x\), vertically stretched by a factor of 2. It has a vertical asymptote at \(x=0\). The domain of \(g(x) = 2\ln x\) is \(x > 0\) and its range is all real numbers.

Step 1: Understanding the Transformation

The transformation on \(f(x) = \ln x\) to become \(g(x) = 2\ln x\) is a vertical stretch by a factor of 2. It means that each y-coordinate of \(f(x) = \ln x\) is scaled by a factor of 2. The shape of the graph remains the same, but y-values are doubled.

Step 2: Determining and Graphing Asymptotes

The original function \(f(x) = \ln x\) has a vertical asymptote at \(x=0\). Since the transformation is a vertical scaling, it will not affect the x-coordinate of the vertical asymptote. Therefore, for \(g(x) = 2\ln x\), there is a vertical asymptote at \(x=0\). Sketch this on the graph.

Step 3: Generating the Graph

Now we apply the transformation on the original graph. The \(y\)-coordinate of every point on the graph needs to be multiplied by 2. So every point \((x, y)\) on \(f(x) = \ln x\) turns into a point \((x, 2y)\) on \(g(x) = 2\ln x\).

Step 4: Define Domain and Range

The domain of \(g(x) = 2\ln x\) is \(x > 0\) and the range is \(-\infty < y < \infty\) which means all real numbers.

Key Concepts

Function Transformations

Function transformations are important when altering or shifting the original graph of a function to produce a new one. Particularly, transformations can include translations, reflections, rotations, and stretches or compressions. When graphing functions like logarithmic functions, transformations are often applied to make the graph better fit data or reflect real-world scenarios.

In the exercise, the standard logarithmic function given by \( f(x) = \ln x \) undergoes a transformation to become \( g(x) = 2\ln x \). This involves a vertical stretch, which means making adjustments to how the function slopes or widens, without changing its horizontal position. Such transformations do not affect the intrinsic shape and properties of the curve itself but modify how steeply it climbs upwards as \( x \) increases.

Understanding these transformations makes it easier to predict how the graph will behave when altered and also assists in sketching functions quickly without needing detailed calculations.

Vertical Stretch

A vertical stretch is a type of transformation that affects the graph of a function by scaling its y-coordinates. It does not alter the x-coordinates, meaning the sideways position remains unchanged.

For the function \( g(x) = 2\ln x \), the vertical stretch is applied by multiplying the \( y \)-values by 2. For example, if a point on the original function is \( (x, y) \), it becomes \( (x, 2y) \) in the transformed function. This results in the graph appearing narrower or more stretched along its vertical axis.

Benefits of using a vertical stretch include:

• Amplifying signal or trend evident in data.
• Increasing sensitivity of model outcomes to input variables.

Understanding how to work with vertical stretches helps in manipulating graphs effectively to model various phenomena accurately.

Asymptotes

An asymptote is a line that a graph approaches but never actually reaches. They can be vertical, horizontal, or oblique, depending on the direction of approximation.

In the basic logarithmic function \( f(x) = \ln x \), a vertical asymptote exists at \( x = 0 \). This is because as \( x \) approaches 0 from the right, the \( f(x) \) values trend towards negative infinity. The transformation to \( g(x) = 2\ln x \) does not move this vertical asymptote since the stretch only affects the y-values.

Vertical asymptotes are crucial elements to consider when plotting graphs since they shape the behavior of a function's graph close to these axes. They help in defining constraints and characteristics of the function, like guiding how we understand continuity and limits in calculus.

Domain and Range

The domain and range of a function describe where the function is defined and what values it can take. The domain is the set of possible input values \( x \), while the range is the collection of potential output values \( y \).

For \( g(x) = 2\ln x \), the domain remains \( x > 0 \), just like its parent function \( \ln x \). This is because the logarithm function is undefined for non-positive values. The range, however, covers all real numbers,\(-\infty < y < \infty \), because the vertical stretching affects only the scale and not the extent of \( y \)-values.

It's essential to note the domain and range when graphing functions, as they determine the scope where the function is valid and its output possible. Identifying these ensures functions are applied correctly in real situations.