Problem 54 - Algebra and Trigonometry

Question

In Exercises \(53-58,\) 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. $$ g(x)=\log _{2}(x+2) $$

Step-by-Step Solution

Answer

The vertical asymptote for the function \(g(x)=\log_{2}(x+2)\) after the transformation is at \(x=-2\). The domain of the function is \((-2, +\infty)\) and the range is \((- \infty, +\infty)\).

Step 1: Graphing base function

Start by sketching the function \(f(x)=\log_{2}x\). This is a standard logarithmic function, so it should look like a curve that passes through the points (1,0) and (2,1) and has a vertical asymptote at \(x=0\). The original domain of \(f(x)\) is \((0, +\infty)\) and the range is \((- \infty, +\infty)\).

Step 2: Understanding the transformation

Now look at the given function \(g(x)=\log_{2}(x+2)\) which is a horizontal shift (specifically, a shift 2 units to the left) of the base function \(f(x)=\log_{2}x\). This means each point on the original function \(f(x)=\log_{2}x\)'s graph will be shifted 2 units left.

Step 3: Graphing the transformed function

By shifting each point of the original graph 2 units to the left, graph the function \(g(x)=\log_{2}(x+2)\). This will seem like 'picking up' the original graph and 'placing it' 2 units to the left. This shift also changes the vertical asymptote to \(x=-2\).

Step 4: Determining the domain and range

Finally, note down the domain of \(g(x)\) which is \((-2, +\infty)\) (due to the shift), and the range is still \((- \infty, +\infty)\), as logarithmic functions always have this range and shifting doesn't affect it.

Key Concepts

Logarithmic Transformations

Logarithmic transformations involve altering the basic shape and position of a logarithm's graph. When you're tasked with graphing functions like g(x) = \(log _{2}(x+2)\), understanding these transformations is key to successfully plotting the graph without having to calculate numerous points.

One common transformation is the horizontal shift, which occurs when a constant is added or subtracted within the logarithmic argument. In our example, the g(x) function includes \(x+2\). This represents a shift 2 units to the left compared to the base function f(x) = \(log_{2}x\). Imagine sliding the entire graph of f(x) to the left; that's exactly what's happening here.

Such transformations follow a general rule: adding a constant inside the argument shifts the graph to the left, and subtracting shifts it to the right. These transformations enable us to efficiently sketch complex logarithmic functions by using the basic logarithm graph as a starting point and making calculated adjustments.

Determining Domain and Range

Determining the domain and range of a function is critical for understanding the extent of its behavior and where its outputs lie. The domain of a logarithmic function, such as f(x) = \(log_{2}x\), is the set of all positive real numbers (\((0, +\infty)\)). This is because you can only take the logarithm of positive numbers.

However, when a logarithmic function undergoes a transformation, its domain shifts accordingly. For the transformed function g(x) = \(log_{2}(x+2)\), the domain becomes \((-2, +\infty)\). This shift is due to the horizontal shift previously mentioned; every value of x in g(x) must be greater than -2 to keep the argument of the logarithm positive.

The range, on the other hand, remains constant for all logarithmic functions with a base greater than 1. It's always \((-\infty, +\infty)\), illustrating that the logarithm can take any real value as its output, regardless of the input or the transformations applied to the function.

Vertical Asymptote

A vertical asymptote of a function is a vertical line that the graph of the function approaches infinitely close to, but never touches or crosses. For the base logarithmic function f(x) = \(log_{2}x\), the vertical asymptote is at \(x=0\) because as x approaches 0 from the right, the outputs increase without bound.

When the function is transformed, for instance, into g(x) = \(log_{2}(x+2)\), the vertical asymptote shifts along with the graph. Therefore, for g(x), the vertical asymptote is no longer at \(x=0\); it moves to \(x=-2\). This new asymptote signifies that the values of g(x) will become increasingly large in magnitude as x approaches -2 from the right, but x can never actually be -2, because the logarithm of zero is undefined.