Problem 74 - 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). $$ g(x)=1-\ln x $$

Step-by-Step Solution

Answer

The graph of the function \(g(x)=1-\ln x\) is formed by applying a vertical shift of 1 to the function \(f(x)=\ln x\). The asymptote is \(y=1\). The domain of \(g(x)\) is \(x>0\), and its range is \(y<1\).

Step 1 - Apply Vertical Transformation

As the function \(g(x)=1-\ln x\), we have to apply a vertical shift to the function \(f(x)=\ln x\). It means that the entire graph of \(f(x)=\ln x\) is moved 1 unit upwards. So, all the y-values are increased by 1.

Step 2 - Drawing the Graph

After we have applied the transformation, we should redraw the graph. The original graph of \(f(x)=\ln x\) has an asymptote at x=0. For \(g(x)=1-\ln x\), this asymptote will also be shifted upwards by 1. So, the graph of \(g(x)\) will approach, but never reach, the horizontal line \(y=1\). Because of the nature of logarithmic function, \(g(x)\) is not defined for \(x \leq 0\). So the curve will remain on the right side of the y-axis.

Step 3 - Asymptote's Equation

As the graph of \(g(x)=1-\ln x\) approaches, but does not reach, the line \(y=1\), this line is a horizontal asymptote of \(g(x)\). Hence, the equation of the asymptote is \(y=1\).

Step 4 - Determine Domain and Range

The domain and range can be determined from the graph. The domain is the set of all possible x-values. Since the function is not defined for negative x-values and zero, the domain of \(g(x)\) is \(x > 0\). The range of a function is the set of all possible y-values. Since the function approaches but never reaches the value 1, the range of the function \(g(x)\) is \(y<1\).

Key Concepts

Vertical Transformations

Understanding vertical transformations can significantly simplify graphing logarithmic functions like the given example, where the function presented is \(g(x)=1-\ln x\). A vertical transformation refers to the shifting of the graph up or down on the Cartesian plane. It's as if you pick up the entire curve and move it vertically without altering its shape. In this case, the graph of the logarithmic function \(f(x)=\ln x\) is moved 1 unit upwards, which results in the function \(g(x)\).

When graphing, each point on the original graph moves straight up by the same amount. As a helpful tip, remember that adding a positive number to a function will shift its graph upwards, and adding a negative number will shift it downwards. This kind of transformation is essential when predicting the behavior of the graph before even plotting it on paper or a screen.

Asymptotes of Logarithmic Functions

Asymptotes play a crucial role in defining the behavior and boundary of logarithmic functions. In the example provided, the function \(g(x)=1-\ln x\) exhibits a horizontal asymptote, which is a horizontal line that the graph approaches but never actually reaches, no matter how far it is extended. The logarithmic function \(f(x)=\ln x\) has an asymptote at \(x=0\). After the vertical transformation, this asymptote is also transformed, resulting in a new horizontal asymptote for the function \(g(x)\).

For \(g(x)=1-\ln x\), the horizontal asymptote would be \(y=1\). No matter how large \(x\) becomes, \(g(x)\) will get closer and closer to the asymptote but will stop short of touching or crossing it. Conclusively, horizontal asymptotes are critical to comprehend since they help to anticipate the long-run behavior of a function's graph.

Domain and Range of Logarithmic Functions

Logarithmic functions have defining characteristics for their domain and range, which are the sets of possible x-values and y-values respectively. For the function \(g(x)=1-\ln x\), we observe that it inherits the domain limitations of its parent function \(f(x)=\ln x\), which cannot take negative values or zero. Consequently, the domain for \(g(x)\) is all positive real numbers, mathematically expressed as \(x > 0\).

In terms of range, since the graph is shifted upwards and the horizontal asymptote for \(g(x)\) is \(y=1\), the function will produce y-values that are less than 1 but never equal to 1, hence the range is \(y<1\). Identifying the domain and range helps in understanding the limitations and extent of a function. Always check for the presence of asymptotes, which may restrict the range, and remember that the logarithm of a number cannot be taken if that number is less than or equal to zero, shaping the domain.