Problem 15 - Contemporary Precalculus

Question

Graph each of the following power functions in a window with \(0 \leq x \leq 20\) (a) \(f(x)=x^{-1.5}\) (b) \(g(x)=x^{75}\) (c) \(h(x)=x^{2.4}\)

Step-by-Step Solution

Answer

Answer: In the given window, the graph of \(f(x)=x^{-1.5}\) is a decreasing function due to its negative exponent. The graph of \(g(x)=x^{75}\) grows very quickly as a result of its large positive exponent. The graph of \(h(x)=x^{2.4}\) is an increasing function and grows more slowly compared to \(g(x)\) due to its relatively smaller positive exponent.

Step 1: (a) Graphing \(f(x)=x^{-1.5}\) in a window \(0 \leq x \leq 20\)

In order to graph this function, we will consider points with increasing values of \(x\) and determine the corresponding value of \(f(x)\). Sketch the points on a graph. Note that when \(x\) is very close to 0, the function will be undefined.

Step 2: (b) Graphing \(g(x)=x^{75}\) in a window \(0 \leq x \leq 20\)

In order to graph this function, we will consider points with increasing values of \(x\) and determine the corresponding value of \(g(x)\). As the exponent is very large, the function will grow very quickly as \(x\) increases. Plot the points on a graph. Remember, the graph begins at \(x=0\).

Step 3: (c) Graphing \(h(x)=x^{2.4}\) in a window \(0 \leq x \leq 20\)

In order to graph this function, we will consider points with increasing values of \(x\) and determine the corresponding value of \(h(x)\). The exponent is positive but smaller than the exponent in \(g(x)\), the function will grow slower than \(g(x)\). Sketch the points on a graph, starting from \(x=0\).

Key Concepts

Graphing Functions

Graphing functions allows us to visualize mathematical relationships and understand how values of functions behave over specific intervals. To start with graphing, it is essential to identify the zero and positive points first, as illustrated in common mathematical functions such as power functions. When graphing power functions like the ones given in the exercise (e.g., \(f(x)=x^{-1.5}\), \(g(x)=x^{75}\), and \(h(x)=x^{2.4}\)), consider these helpful steps:

• Choose a Range: Define a specific range of \(x\)-values, such as \(0 \leq x \leq 20\), to examine how the function behaves.
• Value Calculation: Calculate the function values for selected \(x\) points within that range. This helps in pinpointing key changes or trends in the graph.
• Plot and Connect: Plot these calculated points onto a graph. Then, smoothly connect the points to portray a clearer picture of the function's path.

These steps help to clearly visualize different behaviors of power functions, such as the sharp rise in \(g(x)\), the moderate slow increase in \(h(x)\), and the rapid fall of \(f(x)\) as \(x\) approaches zero.

Exponents

Exponents are crucial in understanding how functions behave. They represent how many times a number, known as the base, is multiplied by itself. In power functions, different exponents indicate different growth rates.For example:

• Negative Exponents: Represent reciprocal values. For \(f(x) = x^{-1.5}\), as \(x\) decreases towards zero, the values of the function increase without bounds.
• Large Positive Exponents: Lead to rapid growth. With \(g(x) = x^{75}\), for any \(x\) greater than 1, the function's value skyrockets quickly.
• Fractional Exponents: As in \(h(x) = x^{2.4}\), suggest a growth rate that is less sharp than whole number exponents but still significant.

Understanding these relationships with exponents helps predict and interpret the behavior of functions more accurately.

Precalculus

Precalculus serves as a foundation for more advanced studies in mathematics. It combines knowledge of algebra and trigonometry to prepare students for calculus. Key topics typically include:

• Function Analysis: Study of different types of functions, like power, exponential, and logarithmic functions, focusing on their properties, transformations, and graphs.
• Understanding Limits: Learning limits helps transition into calculus, offering insights into the behavior of functions as they approach certain points or infinity.
• Function Behavior: Evaluate how specific functions behave when variable values change, specifically when examining limits of the domain or approach undefined values.

By grasping the essentials of precalculus, students are better equipped to tackle the challenges of more complex calculus concepts in their educational journey.