Problem 71 - 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). $$ h(x)=-\ln x $$

Step-by-Step Solution

Answer

The graph of \(h(x)=-\ln x\) is a reflection of \(f(x)=\ln x\) over the x-axis. The vertical asymptote is x=0. The domain of \(h(x)\) is (0, +∞) and the range of \(h(x)\) is (-∞, +∞).

Step 1: Understanding the transformation.

The transformation from \(f(x)\) to \(h(x)\) involves multiplying the y-values by -1. In graphical terms, this transformation will reflect the graph of \(f(x)\) across the x-axis.

Step 2: Implementation of the transformation.

After identifying the transformation, implement it on the graph. The curve which was above the x-axis in \(f(x)\) will now be below the x-axis in \(h(x)\), and vice versa.

Step 3: Identify the asymptotes.

The vertical asymptote of \(f(x)\) is the y-axis (x=0). After the transformation, this does not change, so the vertical asymptote of \(h(x)\) is also x=0.

Step 4: Determine the function's domain and range.

The domain of \(h(x)\) remains the same as \(f(x)\), which is (0, +∞). Since the range of \(f(x)\) was (-∞, +∞), reflecting it over the x-axis will not change the range, so the range of \(h(x)\) is also (-∞, +∞).

Key Concepts

Logarithmic Functions

Logarithmic functions, like \( f(x) = \ln x \), are the inverse of exponential functions. In a logarithmic function, you find the power to which a base number must be raised to get a specific value. This base is typically \( e \), the natural logarithm base, in \( \ln x \).

The graph of \( f(x) = \ln x \) starts at the point \( (1,0) \) and increases to the right, getting closer and closer to negative infinity as it approaches \( x = 0 \).

• Logarithmic graphs are continuous and only defined for \( x > 0 \).
• The curve never touches the y-axis but approaches it increasingly closely, showcasing its asymptotic nature.

Understanding these properties helps us manipulate and transform logarithmic functions to create new graphs.

Reflecting Graphs

Graph transformations often involve reflecting graphs across axes. For the function \( h(x) = -\ln x \), we reflect \( f(x) = \ln x \) across the x-axis.

This reflection transformation changes the sign of every y-value. As a result, points above the x-axis move below it and vice versa. Here’s how it works:

• Point \((1,0)\) remains unchanged because flipping \(0\) does not affect its value.
• Other points, like \((2, \ln 2)\), will be transformed to \((2, -\ln 2)\), reflecting the negative of the original y-value.

By understanding reflection, you can easily sketch \( h(x) \) based on \( f(x) \) without recalculating every point.

Domain and Range

Determining the domain and range is crucial for understanding function behavior. The domain consists of all possible x-values, and the range consists of all possible y-values for a function.

For \( f(x) = \ln x \) and \( h(x) = -\ln x \):

• The domain is \((0, +∞)\). The function is undefined for \( x \leq 0 \).
• The range of \( f(x) \) is \((-∞, +∞)\). Reflecting across the x-axis does not affect this, so \( h(x) \) also has a range of \((-∞, +∞)\).

Identifying domain and range helps us place limits on how the function behaves, ensuring accurate graph transformations and proper function understanding.

Asymptotes

An asymptote is a line that a graph approaches but never touches. In logarithmic functions, identifying asymptotes helps predict long-term behavior.

For \( f(x) = \ln x \) and its reflected graph \( h(x) = -\ln x \):

• The vertical asymptote is \( x = 0 \), indicating the graph gets infinitely close to but never reaches the y-axis.
• This asymptote remains unchanged in transformations like reflection because the shift does not affect \( x \).

Grasping the concept of asymptotes is essential in understanding how the graph behaves as \( x \) approaches the boundary of its domain without actually crossing it.