Problem 5 - Algebra and Trigonometry

Question

\(5-12\) . A function \(f\) is given. (a) Use a graphing device to draw the graph of \(f\) . (b) State approximately the intervals on which \(f\) is increasing and on which \(f\) is decreasing. $$ f(x)=x^{2 / 5} $$

Step-by-Step Solution

Answer

Increasing: \((0, \infty)\); Decreasing: \((- fty, 0)\).

Step 1: Understand the Function

We need to analyze the given function: \( f(x) = x^{2/5} \), which is a power function. This indicates how we would expect the graph \(f\) to behave: flattening out as \( x \) approaches 0, and gradually increasing as \( x \) moves away from 0.

Step 2: Graphing the Function

Use a graphing calculator or software to plot \( f(x) = x^{2/5} \). From the graph, you'll notice that the graph has a smooth curve which both increases for positive \(x\) values and decreases to the left for negative \(x\) values.

Step 3: Determine Intervals of Increase and Decrease

Based on the graph, state the intervals:- \(f\) is increasing on the interval \((0, \infty)\) (positive \(x\) values increase as you move rightwards).- \(f\) is decreasing on the interval \((-fty, 0)\) (negative \(x\) values show a downward trend as you move leftwards).

Key Concepts

Graphing Functions

Graphing functions helps us visually understand how a mathematical expression behaves across different values of its input variable. When we take the function \(f(x) = x^{2/5}\), we look at it as a power function where the exponent dictates its shape on the graph.
To effectively graph a function:

• Use graphing tools such as graphing calculators or software tools—for example, Desmos, GeoGebra, or a TI calculator.
• Plot points to see how they connect on the curve, particularly focusing on key points such as where \(x=0\) or \(x=1\).
• Observe how the function behaves as \(x\) moves towards infinity or negative infinity.
• Note any symmetry or special properties such as asymptotes or roots.

In this exercise, by graphing \(x^{2/5}\), we recognize that it flattens as it approaches zero from both sides and increases without bounds as \(x\) becomes more positive.

Increasing and Decreasing Intervals

Identifying increasing and decreasing intervals in a function helps us understand how the function moves across the graph. An increasing interval is where the function rises as we move from left to right, whereas a decreasing interval is where it falls.
For the function \(f(x) = x^{2/5}\):

• On the interval \((0, \infty)\), \(f(x)\) is increasing. This means that as \(x\) gets larger, \(f(x)\) also gets larger. You can visualize this on the graph as a continual upward slope moving to the right.
• On the interval \((-\infty, 0)\), \(f(x)\) is decreasing. Here, as \(x\) becomes more negative, \(f(x)\) decreases. This corresponds to a downward slope moving to the left on the graph.

Identifying these intervals involves understanding the derivative, but visually, it’s about observing how the graph moves.

Algebraic Functions

Algebraic functions are one of the most fundamental types of mathematical functions, composed using basic algebraic operations: addition, subtraction, multiplication, division, and root extraction. The function \(f(x) = x^{2/5}\) is an example of a root function, which falls under the category of power functions.
Power functions have the form \(f(x) = x^{a/b}\), where \(a/b\) is a rational number. These functions reveal how different powers and roots affect the curvature and behavior of the function.
Understanding the function \(x^{2/5}\):

• The exponent \(2/5\) suggests a root (fifth root in this case) and also a squaring, which influences the graph's flattening behavior near zero.
• Algebraic functions can vary greatly based on their exponents. Rational exponents can result in unique curves that differ significantly from simpler polynomials.

Recognizing the structure of an algebraic function allows us to predict its basic shape and behavior without even plotting it.