Problem 29 - College Algebra Essentials

Question

Begin by graphing \(f(x)=2^{x}\). Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. $$h(x)=2^{x+1}-1$$

Step-by-Step Solution

Answer

The function \(f(x) = 2^x\) has a graph which is always positive and increases as x increases. The function \(h(x) = 2^{x+1} - 1\) is a transformation of \(f(x)\) which shifts every x value to the left by 1 and every y value downwards by 1. The domain of both functions is (-∞, ∞) while the range of \(f(x)\) is (0, ∞) and the range of \(h(x)\) is (-1, ∞).

Step 1: Graph the Base Function

Graph the base function \(f(x) = 2^x\). It is an exponential function, with the y-intercept at (0,1). The function is always positive and increases as x increases. There are no horizontal or vertical asymptotes.

Step 2: Identify the Transformations

Analyzing the function \(h(x) = 2^{x+1} - 1\), it can be observed that it is a transformation of the base function where every x value is decreased by 1 (horizontal shift to the left) and every y value is decreased by 1 (vertical shift downwards).

Step 3: Graph the Transformed Function

Using the transformations identified, graph the function \(h(x) = 2^{x+1} - 1\). The y-intercept shifts to (0,-1) and there's still no horizontal asymptote. However, a horizontal asymptote appears at y = -1 due to vertical shift downwards.

Step 4: Determine the Domain and Range

The domain of \(f(x) = 2^x\) is (-∞, ∞) since the graph extends indefinitely in both directions along the x-axis. The range is (0, ∞) since the function's graph is above the x-axis for all x-values. For the transformed function \(h(x) = 2^{x+1} - 1\), the domain remains (-∞, ∞) but the range changes to (-1, ∞) due to the vertical shift.

Step 5: Confirm with a Graphing Utility

To check the hand-drawn graphs, use a graphing utility to plot both functions. They should match the hand-drawn graphs, which provides a confirmation of the correctness of the results. This step is optional and serves mostly to check the accuracy of the manual calculations.