Problem 36 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the functions \(f, g,\) and \(h\) in the same viewing window. $$f(x)=4-x^{2}, \quad g(x)=x, \quad h(x)=f(x) / g(x)$$

Step-by-Step Solution

Answer

The graphs of functions \(f(x) = 4-x^2\) and \(g(x) = x\) are a downward opening parabola and a straight line with positive slope, respectively. The graph of function \(h(x) = f(x) / g(x)\) depicts the relationship between \(f(x)\) and \(g(x)\) with an exception for \(x=0\) where it is undefined as division by zero occurs.

Step 1: Identify and understand the functions

The given functions are \(f(x)=4-x^{2}\), a downward-opening parabola, \(g(x)=x\), a straight line through the origin with a positive slope, and \(h(x)=f(x) / g(x)\), a rational function where the numerator is \(f(x)\) and the denominator is \(g(x)\).

Step 2: Graph the function \(f(x) = 4 - x^2\)

To graph the function \(f(x) = 4 - x^2\), start by plotting a few points like (-2, 0), (-1, 3), (0, 4), (1, 3), (2, 0), then connect these points smoothly to form a parabola. This graph will face downwards because the coefficient of \(x^2\) is negative.

Step 3: Graph the function \(g(x) = x\)

The function \(g(x) = x\) is a straight line through the origin with a positive slope. You can plot some points like (-2, -2), (-1, -1), (0, 0), (1, 1), (2, 2), and then connect these points with a straight line.

Step 4: Graph the function \(h(x) = f(x) / g(x)\)

To graph the function \(h(x) = f(x) / g(x)\), you divide the value of function \(f(x)\) at each point by the corresponding value of function \(g(x)\). But remember, for \(x=0\), \(h(x)\) is undefined as it leads to division by zero. So exclude this point while plotting for \(h(x)\).

Step 5: Include all three graphs in the same window

Finally, present the graphs of \(f(x)\), \(g(x)\), and \(h(x)\) in the same viewing window. This gives a comparative view of all three functions.

Key Concepts

Downward-Opening Parabola

A downward-opening parabola is a specific type of parabolic graph that curves downwards. The function for a downward-opening parabola is often expressed in the form of \[ f(x) = ax^2 + bx + c \] where the coefficient \(a\) is negative.

• When \(a\) is negative, this indicates that the parabola opens downward.
• At the vertex, the function reaches its maximum value, a distinguishing feature of downward-opening parabolas.
• The graph of \( f(x) = 4 - x^2 \) is a perfect example. It appears as an arch pointing downwards with its vertex at the highest point.

To graph this, plot key points across the parabola and connect them smoothly. In our example, points such as \((-2, 0), (2, 0), (0, 4)\) help reveal the shape of the parabola.

Rational Function

A rational function is a function represented by the ratio of two polynomials.

• The function \( h(x) = \frac{f(x)}{g(x)} \) is an example of a rational function.
• Each term in the function should be simplified as much as possible for easier graphing and analysis.
• Rational functions can exhibit interesting behaviors, such as vertical and horizontal asymptotes.

In the exercise, the function \( h(x) = \frac{4-x^2}{x} \) is constructed by dividing the parabolic function by the linear function. Understanding these functions involves knowing their behavior at different parts of the graph, especially where they are undefined or do not yield real numbers.

Division by Zero

Division by zero is an undefined operation in mathematics, which can lead to important considerations when graphing rational functions.

• This undefined condition arises when trying to divide any number by zero, represented mathematically as: \( \frac{a}{0} \).
• For rational functions, it creates vertical asymptotes or undefined points in the graph.
• In the problem, \( h(x) = \frac{4-x^2}{x} \) becomes undefined at \(x = 0\), creating a point of discontinuity.

When graphing, this means \( h(0) \) should be excluded, and you'll observe a gap on the graph along the vertical line where the denominator equals zero.

Graphing Utility

A graphing utility is a valuable tool used to visualize mathematical functions and their interactions. These tools help in:

• Drawing precise graphs for complex functions with various elements like curves, lines, and gaps efficiently.
• Comparing multiple function graphs simultaneously in the same viewing window, aiding in better understanding.
• Providing visual clarity on concepts like intersections, asymptotes, and undefined regions in functions.

In this exercise, using a graphing utility allows for the overlay of \( f(x) \), \( g(x) \), and \( h(x) \), offering a clear comparative view of the downward-opening parabola, the line, and the rational function, respectively. It's an excellent method to 'see' and verify your analytical work visually.