I appreciate any help I can get on this. I also attached the graphing guidelines :) ### Instructions for Graphing Function on Graph Paper**Using the guidelines found on page 228 to make a complete graph of \( f(x) = x \left(1 - 3x^{-\frac{1}{3}}\right) \).**1. **Preparation**: - Gather graph paper, a pencil, and a ruler.2. **Graphing the Function**: - Begin by plotting key points for the function \( f(x) = x \left(1 - 3x^{-\frac{1}{3}}\right) \). - Pay close attention to critical points, asymptotes, and intercepts. - Plot additional points to ensure the accuracy and smoothness of the graph. - Label axes and scale appropriately.3. **Analyzing the Function**: - Identify and clearly mark the function’s intercepts, where \( f(x) = 0 \). - Determine the function’s behavior (increases, decreases, concavities) by analyzing its first and second derivatives. - Highlight any asymptotic behavior or points of discontinuity. 4. **Applying Theorems**: - When referencing a theorem to derive conclusions about the function’s behavior (e.g., the Intermediate Value Theorem, Mean Value Theorem), ensure that these theorems are cited properly, specifying their exact location in the textbook or other sources.5. **Finalizing the Graph**: - After plotting all necessary points and behaviors, connect them smoothly to form the complete graph. - Ensure the graph is neat and all key features (intercepts, critical points, asymptotes, etc.) are clearly labeled. By following these steps, a thorough and accurate graph of \( f(x) = x \left(1 - 3x^{-\frac{1}{3}}\right) \) will be created on graph paper. This exercise will reinforce understanding of graphing complex functions manually and applying mathematical theorems to analyze their properties. ### Graphing Guidelines for \( y = f(x) \)1. **Identify the domain or interval of interest.** - Determine the interval(s) on which the function should be graphed. It may be the domain of the function or some subset of the domain.2. **Exploit symmetry.** - Take advantage of symmetry. For instance, assess if the function is even (\( f(-x) = f(x) \)), odd (\( f(-x) = -f(x) \)), or neither.3. **Find the first and second derivatives.** - These are needed to determine extreme values, concavity, inflection points, and the intervals on which \( f \) is increasing or decreasing. Computing derivatives—particularly second derivatives—may not always be practical, so some functions may need to be graphed without complete derivative information.4. **Find critical points and possible inflection points.** - Determine points at which \( f'(x) = 0 \) or \( f' \) is undefined. Identify points at which \( f''(x) = 0 \) or \( f'' \) is undefined.5. **Find intervals on which the function is increasing/decreasing and concave up/down.** - The first derivative determines the intervals on which \( f \) is increasing or decreasing. The second derivative determines the intervals on which the function is concave up or concave down.6. **Identify extreme values and inflection points.** - Use either the First or Second Derivative Test to classify the critical points. Both \( x \)- and \( y \)-coordinates of maxima, minima, and inflection points are needed for graphing.7. **Locate all asymptotes and determine end behavior.** - Vertical asymptotes often occur at zeros of denominators. Horizontal asymptotes require examining limits as \( x \to \pm \infty \); these limits determine end behavior. Slant asymptotes occur with rational functions in which the degree of the numerator is one more than the degree of the denominator.8. **Find the intercepts.** - The \( y \)-intercept of the graph is found by setting \( x = 0 \). The \( x \)-intercepts are found by solving \( f(x) = 0 \); these are the real zeros (or roots) of \( f \).9.

Explanation: Given that, 1. Find the Domain. Here we see…

Graphing Guidelines for y = f(x) 1. Identify the domain or interval of interest. On what interval(s) should the function be graphed? It may be the domain of the function or some subset of the domain. 2. Exploit symmetry. Take advantage of symmetry. For example, is the function even ( f(-x) = f(x)), odd (f(-x) = -f(x)). or neither? 3. Find the first and second derivatives. They are needed to determine extreme values, concavity, inflection points, and