Problem 34 - Precalculus with Limits

Question

GRAPHICAL REASONING In Exercises 33-36, 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 \(x > 6\)? \(f(x) = \frac{x}{2}\), \(g(x) = \sqrt{x}\)

Step-by-Step Solution

Answer

The function that contributes the most to the magnitude of the sum in each interval can be determined visually by creating the graph and comparing the functions in each interval. The function with the higher y-coordinates in a given interval contributes more to the sum magnitude in that interval.

Step 1: Graph the Functions

Use a graphing utility to plot the functions \(f(x) = \frac{x}{2}\) and \(g(x) = \sqrt{x}\), as well as the result of their sum \(f(x) + g(x)\), on the same graph. The x-axis should cover the range according to the problem, i.e. from \(0\) to a value greater than \(6\).

Step 2: Compare Magnitudes in Interval \(0 \leq x \leq 2\)

Now, investigate the interval \(0 \leq x \leq 2\). The magnitude of the sum function \(f+g\) is determined by the function that has higher y-coordinates in this interval. Hence, the function that has generally larger y-coordinates in this interval contributes more to the magnitude of the sum.

Step 3: Compare Magnitudes in Interval \(x > 6\)

Next, focus on the interval \(x > 6\). Similarly to the previous step, analyse which function, \(f\) or \(g\), contributes more to the magnitude of the function sum \(f+g\) in this interval based on the comparison of their y-coordinates.

Key Concepts

Graphing Utility

When tackling precalculus problems, a graphing utility is an invaluable tool for visualizing functions and their relationships. Working with the functions f(x) = \frac{x}{2} and g(x) = \sqrt{x} using a graphing utility can make it easier to comprehend their behaviors over certain intervals. By plotting these functions on the same set of axes, we gain insight into how they change and interact.

Both linear and square root functions have distinct shapes and rates of growth. In the exercise, overlaying the sum of the functions f(x) + g(x) provides a merged visual that illustrates which function has more influence over the combined output at different points on the x-axis. Use the zoom function to focus on key intervals like \(0 \leq x \leq 2\) and \(x > 6\), making it easier to see which function dominates in each range and enhancing the learning experience with clear graphical evidence.

Function Magnitude

The magnitude of a function, commonly referred to as its absolute value, represents the function's size regardless of its direction. When comparing the magnitude of two functions within a specific interval, you're essentially asking which function's values are larger in that interval. In our example with f(x) and g(x), you'll compare their outputs (y-values) over the intervals given in the exercise.

It's crucial to identify where on the graph the functions have their greater y-coordinates in the first interval, \(0 \leq x \leq 2\), and similarly, for the second interval \(x > 6\). Remember, when comparing function magnitudes, it's the height of the function above the x-axis that counts - the physical distance from the axis, irrespective of whether the function's values are positive or negative.

Sum of Functions

The sum of functions is a combined function, the result of adding the output values of two separate functions for the same input value of x. For example, the sum function f(x) + g(x) adds the values of f and g at each x position to form a new function. This new function can take on characteristics of both contributing functions or might predominantly reflect the more 'powerful' function in the sum.

Understanding this concept involves another layer of analysis. By graphing the sum function alongside its component functions, students can see where the sum is closer to f(x) or g(x), thereby determining which function has greater influence in the sum. This comparative approach not only demonstrates the concept of function addition but also extends the student's understanding of function behavior and interaction within different intervals of the domain.