Problem 52 - 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. $$f(x)=\left(\frac{1}{2}\right)^{x} \text { and } g(x)=\left(\frac{1}{2}\right)^{x-1}+2$$

Step-by-Step Solution

Answer

The asymptote for \(f(x)\) is at y=0 and for \(g(x)\) it's at y=2.

Step 1: Identify the equation form and key features

First, let's identify the two functions we are dealing with: \(f(x)=(1/2)^x\) and \(g(x)=(1/2)^{x-1}+2\). Both of these functions are exponential decays. Here, the base is 1/2 for both. For \(f(x)\), the asymptote will be at y=0. For \(g(x)\) it's a bit trickier, since the function is a vertical shift of \(f(x)\), more specifically, it's shifted up by 2 units. So the asymptote will be at y=2.

Step 2: Graph \(f(x)\) and identify its asymptote

Now, let's graph \(f(x)\) first. Start by plotting points for a small range of x values, like, -2, -1, 0, 1, 2. Sketch the curve smoothly, ensuring that as x approaches infinity, y gets closer and closer to 0 but never quite reaches it. This line, y=0, is the asymptote of \(f(x)\). As a check, note that the plot of \(f(x)\) should be decreasing because \(f(x)\) is an exponential decay function.

Step 3: Graph \(g(x)\) and identify its asymptote

Now, do the same with \(g(x)\). Keep in mind, that \(g(x)\) is the same as \(f(x)\), only shifted up 2 units. So sketch the plot of \(g(x)\) 2 units above the plot of \(f(x)\), meaning the asymptote will be 2 units above as well, at y=2. Again, plot points for a small range of x values to guide you.

Step 4: Confirm with a graphing utility

Finally, it is advisable to confirm your hand-drawn graphs with a graphing utility to ensure accuracy.