Problem 25 - 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. $$g(x)=2^{x+1}$$

Step-by-Step Solution

Answer

The graph of \(f(x) = 2^{x}\) is a rising curve passing through the point (0,1). The graph of \(g(x) = 2^{x+1}\) is similarly a rising curve translated one unit to the left. Both functions have a domain of \(\(-∞, +∞\)\) and a range of \((0, +∞)\). Both also have a horizontal asymptote at \(y=0\).

Step 1: Graph the original function

Graph \(f(x)=2^{x}\). This is a basic exponential function and the graph is a rising curve that passes through the point (0, 1) in the second Quadrant.

Step 2: Graph the transformed function

Graph the function \(g(x)=2^{x+1}\). Notice that the transformation here is shifting the graph of the original function one unit to the left. This transformation is obtained by replacing \[x\] in \(f(x)\) with \[(x+1)\].

Step 3: Find the asymptotes

The horizontal asymptote of the original function \(f(x) = 2^{x}\) is \(y = 0\). The function \(g(x) = 2^{x+1}\) also has a horizontal asymptote of \(y = 0\) since vertical shifts do not affect horizontal asymptotes.

Step 4: Determine the Domain and Range

For both functions \(f(x) = 2^{x}\) and \(g(x) = 2^{x+1}\), the domain is \(\(-∞, + ∞\)\) and the range is \((0, +∞)\). This is clear from the graph of the functions.

Step 5: Confirm with a graphing utility

Use a graphing utility to confirm the hand-drawn graphs. The functions should accurately reflect the described behavior.