Problem 59 - Contemporary Precalculus

Question

Sketch the graph of the function. $$h(x)=-2 \log x$$

Step-by-Step Solution

Answer

Answer: The x-intercept is at the point (1, 0).

Step 1: Identify key features of the function

First, determine the domain and range of the given function. Since the logarithmic function has a domain of $$x>0$$, the domain of $$h(x)$$ is also $$x>0$$. The range of a logarithmic function is all real numbers, so the range of $$h(x)$$ is $$(-\infty, \infty)$$.

Step 2: Locate points of interest

Now let's find any intercepts and asymptotes. The graph of $$h(x)$$ will have a vertical asymptote at $$x = 0$$ because logarithmic functions are undefined at 0. To find any x-intercepts, set $$h(x) = 0$$ and solve for x: $$-2 \log x = 0$$ $$\log x = 0$$ $$x = 10^0 = 1$$ Thus, the x-intercept is at the point (1, 0).

Step 3: Determine the general shape of the graph

The graph of $$h(x) = -2 \log x$$ is a vertical stretch by a factor of 2 and a reflection across the x-axis of the parent function $$y = \log x$$. This means that the graph will be more "compressed" vertically and "flipped" across the x-axis compared to the parent function. With these steps, we now have enough information to sketch the graph of $$h(x) = -2 \log x$$. To do this, plot the x-intercept at (1, 0), show a vertical asymptote at $$x = 0$$, and draw the transformed logarithmic function with the appropriate shape.

Key Concepts

Domain and Range of Logarithmic Functions

Understanding the domain and range of logarithmic functions can initially seem daunting, but let's break it down. The domain of a function refers to all the possible input values (x-values) that will produce a valid output. For logarithmic functions such as the given function in our exercise, \( h(x) = -2 \log x \), the domain is limited to positive real numbers. This is because the logarithm of a non-positive number is undefined. Hence, the domain of our function \( h(x) \) is \( x > 0 \) or \( (0, \infty) \) in interval notation.

The range is somewhat simpler, as it comprises all of the potential outputs (y-values) that the function can provide. Logarithmic functions are known for their flexibility in this sense because they can produce any real number as an output. Therefore, the range of \( h(x) \) is all real numbers, or \( (-\infty, \infty) \) in interval notation. Comprehending these characteristics of domain and range will enable students to more confidently sketch and analyze logarithmic functions.

Logarithmic Intercepts and Asymptotes

Intercepts and asymptotes are critical in understanding the behavior of logarithmic functions. To find the x-intercept of our function \( h(x) = -2 \log x \), we set the function equal to zero and solve for x. Doing this, as shown in the step-by-step solution, leads us to the point (1, 0), providing us with an essential anchor point on the graph.

As for asymptotes, they're lines that the graph of the function approaches but never touches. Logarithmic functions feature a vertical asymptote at \( x = 0 \), which is where the function is undefined. This means our graph will approach this vertical line but will never cross it, which is critical when sketching the function as it informs us about the direction and behavior of the curve as it moves along the x-axis.

Transformations of Logarithmic Functions

Transformations can significantly change the appearance of a function's graph. With our function \( h(x) = -2 \log x \), we notice two transformations from the parent function \( y = \log x \): a vertical stretch by a factor of 2, and a reflection across the x-axis. The vertical stretch means that the graph will be more 'compressed' vertically, with each point being twice as far from the x-axis. The reflection flips the graph, which for logarithmic functions means that what would normally be an increasing curve (from left to right) will be decreasing instead. Together these transformations give a distinct shape to our logarithmic function, making understanding transformations integral to accurately graphing functions like \( h(x) \).