Problem 38 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph \(f, g,\) and \(f+g\) in the same viewing window. Which function contributes most to the magnitude of the sum when \(0 \leq x \leq 2 ?\) Which function contributes most to the magnitude of the sum when \(x>6 ?\) $$f(x)=\frac{x}{2}, \quad g(x)=\sqrt{x}$$

Step-by-Step Solution

Answer

For the interval [0,2], the function \(g(x)\) contributes most to the magnitude of the sum. For \(x>6\), the function \(f(x)\) contributes most to the magnitude of the sum.

Step 1: Graph the functions

Graph the functions \(f(x) = \frac{x}{2}\) , \(g(x) = \sqrt{x}\) and \(f(x) + g(x)\) on the same viewing window. For instance, using an online graphing utility, plot these functions. This will visually assist in understanding the relationship between the functions.

Step 2: Examine the intervals [0,2]

In the interval [0,2], inspect the graphs and observe which function \(f(x)\) or \(g(x)\) had a higher magnitude. Remember that magnitude refers to the absolute value of a quantity so we are considering which function has the greater absolute value over this range. The graph would clearly show which function between \(f(x)\) and \(g(x)\) is contributing more to the magnitude of the sum in the given interval.

Step 3: Examine the interval (6, infinity)

Similarly, for \(x>6\), inspect the graph and find out which function has a greater magnitude beyond x=6. again looking at the graph, it would become clear that as x moves beyond 6, which function \(f(x)\) or \(g(x)\) contributes more to the magnitude of the sum in the given interval.