Problem 40 - 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 \(x>6 ?\) $$f(x)=x^{2}-\frac{1}{2}, \quad g(x)=-3 x^{2}-1$$

Step-by-Step Solution

Answer

For \(0 \leq x \leq 2\), the function \(f(x) = x^{2} - 1/2\) contributes most to the magnitude of the sum, and for \(x > 6\), the function \(g(x) = -3x^{2} - 1\) contributes most.

Step 1: Graph Individual Functions

First, graph \(f(x)\) and \(g(x)\) individually on the same graph. Plot \(f(x) = x^{2} - 1/2\) and \(g(x) = -3x^{2} - 1\) and label them accordingly.

Step 2: Graph Sum of Functions

Next, graph the sum of the two functions, i.e., \(f + g\). This is found by adding the value of \(f(x)\) to \(g(x)\) for every \(x\) value. This will give a new function, which should also be plotted on the same graph.

Step 3: Identify Contributions: \(0 \leq x \leq 2\)

Identify which function contributes most to the magnitude of the sum in the given range of \(0 \leq x \leq 2\). This can be done by comparing the graphs of \(f(x)\), \(g(x)\), and \(f + g\) within the specified interval. Observe which function has higher absolute values.

Step 4: Identify Contributions: \(x > 6\)

Lastly, identify which function contributes most to the magnitude of the sum when \(x > 6\). Again, this is done by comparing the magnitudes of the functions and the combined graph within this new interval.