Problem 50 - College Algebra Essentials

Question

Graph functions \(f\) and \(g\) in the same rectangular coordinate system. Graph and give equations of all asymptotes. If applicable, use a graphing utility to confirm your hand-drawn graphs. $$f(x)=3^{x} \text { and } g(x)=3 \cdot 3^{x}$$

Step-by-Step Solution

Answer

Both functions \(f(x) = 3^{x}\) and \(g(x) = 3 \cdot 3^{x}\) have a horizontal asymptote at \(y=0\). The function \(f(x)\) is a standard positive exponential curve, while \(g(x)\) is a similar curve but steeper due to the scaling factor of 3.

Step 1: Identify the Asymptotes

For an exponential function, the horizontal asymptote is always at \(y=0\). So both functions \(f(x)\) and \(g(x)\) have horizontal asymptotes at \(y=0\). They do not have vertical asymptotes.

Step 2: Plot the Graph of \(f(x)\)

Start by plotting a few points for \(f(x)=3^{x}\). Some straightforward points are at \(x=-1, 0, 1\), which gives us the points \((-1, 1/3), (0, 1), (1, 3)\). Connecting these points with a curve, we get the graph of \(f(x)\). This function starts at a small positive value for negative \(x\) (approaching the asymptote), increases to 1 at \(x=0\), and then increases rapidly for positive \(x\).

Step 3: Plot the Graph of \(g(x)\)

For the function \(g(x) = 3 \cdot 3^{x}\), this is identical to the graph of \(f(x)\), scaled up by a factor of 3. So the graph of \(g(x)\) will be steeper, but have the same overall shape.

Step 4: Confirm the Graphs Using a Graphing Utility

Lastly, confirm these hand-drawn graphs using a graphing utility. Both graphs should look like steeper and standard positive exponential curves, respectively, approaching but never reaching the horizontal asymptote at \(y=0\).