Problem 48 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Sketch the graph of the function, using the curve-sketching quide of this section. $$ f(x)=\sqrt[3]{x^{2}} $$

Step-by-Step Solution

Answer

The graph of the function \(f(x) = \sqrt[3]{x^2}\) has a domain of \(-\infty < x < \infty\) and a range of \(0 \leq y < \infty\). The function is even and symmetrical around the y-axis. There are no maxima or minima, but there is an inflection point at x = 0. The function increases without bound as x approaches positive or negative infinity. To sketch the graph, plot the origin and reflect the curve symmetrically across the y-axis.

Step 1: 1. Identify the domain and range

Since the function \(f(x) = \sqrt[3]{x^2}\) is defined for all real numbers, the domain is \(-\infty < x < \infty\). The range of the function is all non-negative real numbers, since we are taking the cube root of a squared value it will always be non-negative, thus the range is \(0 \leq y < \infty\).

Step 2: 2. Determine the symmetry

Since \(f(-x) = \sqrt[3]{(-x)^2} = \sqrt[3]{x^2} = f(x)\), the function is even and symmetrical around the y-axis. This means that the graph of the function is mirror-reflected over the y-axis.

Step 3: 3. Find the critical points

The critical points occur when the first derivative of the function equals zero or does not exist. Let's find the first derivative of the function: \(f'(x) = \frac{d}{dx} \sqrt[3]{x^2}\) Using the chain rule, we have: \(f'(x) = \frac{2}{3}x^{-\frac{1}{3}}\) Now, the first derivative does not exist at x = 0, since the exponent is negative. So, the function has no maxima or minima, but x = 0 is an inflection point where the curve changes concavity.

Step 4: 4. Determine the behavior as x approaches infinity

As \(x \to \pm\infty\), the function \(f(x) = \sqrt[3]{x^2}\) also goes to infinity, since we are dealing with the cube root of a squared number. Thus, the function increases without bound in both directions.

Step 5: 5. Plot the function

Using the information gathered in steps 1-4, we can now sketch the graph of the function: - The domain is all real numbers, and the range is all non-negative real numbers. - The function is symmetrical around the y-axis due to its even nature. - There is no maxima or minima, but there is an inflection point at x = 0. - The function increases without bound as x approaches positive or negative infinity. Considering all these facts, you can now sketch the graph of the function \(f(x) = \sqrt[3]{x^2}\). Remember that it should be symmetric, touch the x-axis at the origin, and increase without bound in both the positive and negative x directions.

Key Concepts

Domain and Range

The domain of a function is the complete set of possible input values (x-values), while the range is the set of possible output values (y-values). For the function \( f(x) = \sqrt[3]{x^2} \), we need to determine both. The cube root function is defined for all real numbers, as it can handle any value of \( x \), whether positive, negative, or zero. This means the domain is all real numbers, represented as \( (-\infty, \infty) \).
When it comes to range, because we are taking the cube root of a squared number, the output is always non-negative. The lowest value of \( y \) is 0 (when \( x = 0 \)), and it increases indefinitely as \( x \) becomes larger in the positive or negative direction. Therefore, the range is \( [0, \infty) \).

• Domain: \( (-\infty, \infty) \)
• Range: \( [0, \infty) \)

Symmetry of Functions

Understanding symmetry can simplify the sketching of functions. A function is even if \( f(-x) = f(x) \). This implies that the graph is symmetrical around the y-axis.

The function \( f(x) = \sqrt[3]{x^2} \) is even, as \( f(-x) = \sqrt[3]{(-x)^2} = \sqrt[3]{x^2} = f(x) \). As a result, any point \((x, y)\) on the graph has a corresponding point \((-x, y)\) on the opposite side of the y-axis. This evenness and symmetry can help predict parts of the graph without calculating every point.

• Function is even: \( f(-x) = f(x) \)
• Symmetrical about the y-axis

Critical Points

Critical points occur where the derivative is zero or undefined. These points often signal where the graph of a function changes direction or slope, such as at peaks or troughs.

For \( f(x) = \sqrt[3]{x^2} \), the first derivative is \( f'(x) = \frac{2}{3}x^{-\frac{1}{3}} \). The derivative doesn't exist at \( x = 0 \) because you cannot have a term with a negative exponent equal zero. Hence, there are no local maxima or minima, but this points to a special place in our graph.

• Derivative: \( f'(x) = \frac{2}{3}x^{-\frac{1}{3}} \)
• Critical point at \( x = 0 \) (not differentiable)

Inflection Points

Inflection points are where the graph changes its concave nature from concave up to concave down or vice versa.

Since the derivative of our function \( f'(x) \) changes behavior at \( x = 0 \) (changing from not steep to steep), the function has an inflection point at \( x = 0 \). This is where the graph will shift its curvature, creating a smooth-transformed point.

This single inflection point at the origin gives a clue about the graph's further behavior beyond local slopes.

• Inflection point at \( x = 0 \)