Problem 64 - College Algebra

Question

The figure shows the graph of \(f(x)=\log x .\) 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). $$ g(x)=2-\log x $$

Step-by-Step Solution

Answer

For the function \(g(x) = 2 - \log(x)\), its graph is a vertical shift upwards by 2 units of the graph of the function \(\log(x)\). It has a vertical asymptote at x=0. The domain of the function is (0, +∞) and the range is all real numbers (-∞, +∞).

Step 1: Identify the Original Function and its Transformation

The original function is \(f(x) = \log(x)\). The function \(g(x) = 2 - \log(x)\) is a transformation of the original function\. Here, the function has been shifted upwards by 2 units. This transformation is represented by the \(2-\) in \(g(x) = 2 - \log(x)\).

Step 2: Graph the Transformed Function

To graph the transformed function \(g(x) = 2 - \log(x)\), start with the graph of the original function, \(f(x) = \log(x)\), and shift every point upwards by 2 units. Since this shift does not affect the x-values of the graph, there is still a vertical asymptote at x=0.

Step 3: Find the Domain and Range

The domain of a function is the set of all possible x-values, and the range is the set of all possible y-values on the graph. For the function \(g(x) = 2 - \log(x)\), the domain is the same as the original function \(\log(x)\), which is (0, +∞). The range of the function is all real numbers (-∞, +∞), since there is no upper limit to the y-values due to the shift upwards.

Step 4: Identify the Asymptotes

For the function \(g(x) = 2 - \log(x)\), as with \(f(x) = \log(x)\), there is a vertical asymptote at x=0. This is because as x approaches 0 from the right, the values of \(g(x)\) become very large in the negative direction.

Key Concepts

Graph Transformations

Graph transformations modify the appearance of the original function's graph, such as shifting, stretching, or reflecting. For logarithmic functions like \(f(x) = \log x\), these transformations are crucial to understand how the graph changes. In the given function \(g(x) = 2 - \log x\), a key transformation applied is a vertical shift by 2 units upwards. This happens because the number 2 is added to the function's output, adjusting all y-values by adding 2.

• When a function has the form \(g(x) = a + f(x)\), if \(a > 0\), the graph shifts upwards.
• If \(a < 0\), it shifts downwards.

In this case, the log function hasn’t been horizontally shifted because there's no change in its input \(x\), only a change in its output \(\log x\). As such, the shape of the graph will remain a typical logarithmic curve, just positioned higher on the y-axis. Graph transformations like these help us predict how a graph will look without necessarily needing to plot each point individually.

Domain and Range

Understanding the domain and range of logarithmic functions is essential. For the function \(f(x) = \log x\), the domain is \((0, +\infty)\), meaning that it only accepts positive values of \x\. This characteristic is due to the logarithm being undefined at zero or for any negative number.For the transformed function \(g(x) = 2 - \log x\), the domain remains the same at \((0, +\infty)\). No horizontal transformations are applied, so \x\ still must be positive.

• Domain: \((0, +\infty)\)
• Range: All real numbers \((-\infty, +\infty)\)

The range, however, changes due to the transformation. Originally, \(\log x\) increases to very high values as \x\ increases, creating a range that doesn’t extend below 0. In \(g(x) = 2 - \log x\), subtracting the logarithm from 2 reverses this effect. Now, the values extend in both directions with no restriction, covering all real numbers. Observing how domain and range interact with transformations provides deeper insights into function behavior.

Vertical Asymptotes

Vertical asymptotes are lines that the graph approaches but never actually reaches. They are critical in graphing and understanding the behavior of functions, especially logarithmic ones. For the core function \(f(x) = \log x\), a vertical asymptote exists at \(x = 0\). This is because the function becomes undefined as \x\ approaches zero from the right side. At this vertical line, the function’s values head towards negative infinity, indicating that \(\log x\) never crosses or touches \(x = 0\).In the transformed function \(g(x) = 2 - \log x\), the vertical asymptote remains at \(x = 0\).

• Location: \(x = 0\)
• Behavior: As \x\ approaches 0, \g(x)\ trend to negative infinity.

This means any vertical transformations would need to affect the \(x\) values directly to shift or change the asymptote. Recognizing that the vertical asymptote for \(\log x\) does not shift unless the \(x\) values themselves are transformed helps in accurately graphing and comprehending the behavior and limitations of these functions.