Problem 56 - College Algebra Essentials

Question

Begin by graphing \(f(x)=\log _{2} x .\) Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. $$h(x)=2+\log _{2} x$$

Step-by-Step Solution

Answer

The graph of the function \(h(x) = 2 + log_2x\) can be obtained by shifting the graph of the base function \(f(x) = log_2x\) upwards by 2 units. The vertical asymptote is \(x=0\). The domain of both functions is \(x > 0\), while the range of \(f(x) = log_2x\) is \(y > 0\) and the range of \(h(x) = 2 + log_2x\) is \(y > 2\).

Step 1: Understand and plot \(f(x) = log_2x\)

To plot the graph of \(f(x) = log_2x\), start by noting that this function is defined for \(x > 0\). For basic values, when \(x=1\), \(y=log_21=0\), and when \(x=2\), \(y=log_22=1\). The graph is increasing and has a vertical asymptote at \(x=0\).

Step 2: Apply Transformations to the Function

The function \(h(x) = 2 + log_2x\) is a transformation of the function \(f(x) = log_2x\). Specifically, the '+2' shifts the graph vertically upward by 2 units. So, taking every point on the original graph \(f(x) = log_2x\) and shifting it up 2 units will create the graph of \(h(x) = 2 + log_2x\).

Step 3: Determine the Vertical Asymptote

The vertical asymptote of the basic logarithmic function \(log_2x\) is \(x=0\). When applying the upwards shift, the position of the vertical asymptote does not change. So, the vertical asymptote of \(h(x) = 2 + log_2x\) also remains at \(x=0\).

Step 4: Find Domain and Range

The domain of a logarithmic function is the set of positive real numbers. So, the domain of both \(f(x) = log_2x\) and \(h(x) = 2 + log_2x\) is \(x > 0\). The range of a logarithmic function is usually the set of all real numbers, but for \(h(x) = 2 + log_2x\), due to the upwards shift of 2 units, the range will now be \(y > 2\) instead of \(y > 0\).

Key Concepts

Logarithmic Transformations

When studying logarithmic functions, understanding how basic graphs can be altered, or transformed, is crucial. Logarithmic transformations involve procedures that change the appearance of the graph without altering the function's basic nature. For instance, take the function given as an exercise, f(x) = \(\log_2 x\). This function represents the parent graph from which we can derive other forms through transformations.

Now, by adding 2 to the function, h(x) = 2 + \(\log_2 x\), we are introducing a vertical shift. Here's what occurs: Every point on the graph of f(x) rises two units upwards, creating a new graph for h(x). This transformation doesn’t affect the x-values but adds 2 to all the y-values. It's essential to carry out these transformations precisely, using the parent function as a baseline from which the new function is plotted.

Vertical Asymptote

In the realm of graphing logarithms, the vertical asymptote plays a significant role. It's a vertical line to which the graph of a function approaches as the input either increases towards positive infinity or decreases towards zero. For a basic logarithmic function like f(x) = \(\log_2 x\), this line is located at x = 0. No matter how x approaches zero, the output y races off to infinity, but the curve will never actually cross this invisible boundary.

In applied transformations such as h(x) = 2 + \(\log_2 x\), despite the graph shifting vertically, the vertical asymptote remains at x = 0. This unchangeable property ensures that no horizontal or vertical shift can alter the position of the vertical asymptote for logarithmic functions. It serves as a constant reminder of the behavior and restrictions of these types of functions.

Domain and Range of Logarithmic Functions

The domain and range are two concepts that define the set of possible input values (domain) and the resulting output values (range) for a function. Specifically for logarithmic functions like f(x) = \(\log_2 x\) and the transformed h(x) = 2 + \(\log_2 x\), this remains a pivotal part of understanding how these functions behave.

The domain of logarithmic functions is always x > 0, because the logarithm of a non-positive number is undefined. It corresponds to the idea that you cannot find a real number exponent that will raise a base to a negative or zero result. On the other hand, the range of the basic logarithmic function f(x) is all real numbers, as theoretically, the log function can output any value from negative to positive infinity. However, with our transformed function h(x), the entire graph shifts upwards, thus the range becomes y > 2 rather than extending indefinitely both above and below the x-axis. Noting these characteristics is essential when graphing these functions and predicting their behaviors.