Problem 54 - 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. $$g(x)=\log _{2}(x+2)$$

Step-by-Step Solution

Answer

The vertical asymptote for the function \(g(x)=\log _{2}(x+2)\) is at x = -2. The domain of \(g(x)\) is (-2, +∞) and the range is (-∞, +∞).

Step 1: Graph the Original Function

Start by graphing the original function \(f(x)=\log _{2} x\). Recall that the graph of this function will start at (1,0), because \(\log_2(1) = 0\), and curve upwards towards the right. Also note that the function is undefined for x ≤ 0 due to the nature of logarithms, and therefore, the vertical asymptote is at x=0.

Step 2: Perform the Transformation

Now, to graph the function \(g(x)=\log _{2}(x+2)\), realize that this function is a horizontal shift of \(f(x)=\log _{2} x\) to the left by 2 units. All points on the graph will move 2 units to the left. This new function will still curve upward towards the right, but now it will start at (-1,0), because \(\log_2(-1+2) = 0\). The vertical asymptote now will be moved to x=-2 due to this shift.

Step 3: Determine the Domain and Range

The domain of a function is the set of all possible x-values, while the range is the set of all possible y-values. For \(g(x)=\log _{2}(x+2)\), since the function is now defined for all x > -2, the domain will be (-2, +∞). Since the function values will range from negative to positive infinity, the range will be (-∞, +∞).