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Chapter 2: Problem 36

A function \(f\) is given. (a) Use a graphing calculator to draw the graph of \(f .\) (b) Find the domain and range of \(f .\) (c) State approximately the intervals on which \(f\) is increasing and on which \(f\) is decreasing. $$f(x)=x^{3}-4 x$$

Short Answer

Expert verified
The domain and range of \(f\) are all real numbers. \(f\) increases on \((-1.15, 1.15)\) and decreases on \((-\infty, -1.15)\) and \((1.15, \infty)\).

Step by step solution

01

Graph the Function

Use a graphing tool to plot the function \(f(x) = x^3 - 4x\). Observe the shape of the graph to understand the function's behavior. The plot should help visualize where the function increases or decreases.
02

Determine the Domain

The domain of a polynomial function like \(f(x) = x^3 - 4x\) is all real numbers \((-fty, fty)\) because there are no restrictions on the values \(x\) can take.
03

Determine the Range

To determine the range of \(f(x) = x^3 - 4x\), consider that as \(x\) becomes very large or very small, \(x^3\) will dominate \(-4x\), causing the function to tend towards both \(fty\) and \(-fty\). Hence, the range is also all real numbers \((-fty, fty)\).
04

Find Critical Points

Find the derivative of \(f(x)\), \(f'(x) = 3x^2 - 4\). Set the derivative to zero to find critical points: \(3x^2 - 4 = 0\). Solve for \(x\): \(x = \pm \sqrt{\frac{4}{3}}\). These points are approximately \(x = \pm 1.15\).
05

Test Intervals Around Critical Points

Test the intervals \((-\infty, -1.15)\), \((-1.15, 1.15)\), and \((1.15, \infty)\) by choosing test points in each interval and determine if \(f'(x)\) is positive (function increasing) or negative (function decreasing).
06

Identify Increasing and Decreasing Intervals

For \((-\infty, -1.15)\), \(f'(x) < 0\) (decreasing); for \((-1.15, 1.15)\), \(f'(x) > 0\) (increasing); and for \((1.15, \infty)\), \(f'(x) < 0\) (decreasing). Therefore, the function decreases from \(-\infty\) to \(-1.15\), increases from \(-1.15\) to \(1.15\), and then decreases again from \(1.15\) to \(\infty\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Calculator
A graphing calculator is a beautiful tool that brings equations to life, allowing us to visualize functions easily. To graph the function \( f(x) = x^3 - 4x \), input the equation into your graphing calculator. The graph will help you see the shape and the overall behavior of the function.
  • Use the zoom feature to get a closer look at key features like intercepts and turning points.
  • By adjusting the window settings, you can focus on specific intervals or explore broader areas of the graph.
  • This visual representation makes it easier to understand where the function increases, decreases, and how it approaches infinity.
Experimenting with different functions on a graphing calculator helps you build intuition about their behavior. It's a hands-on way of learning that can enhance your understanding of calculus concepts significantly.
Domain and Range
The domain and range of a function describe its input and output possibilities. For polynomial functions like \( f(x) = x^3 - 4x \), understanding these concepts is straightforward.
  • Domain: Because polynomial functions are continuous and smooth, their domain is all real numbers. There are no breaks, gaps, or undefined points. Thus, the domain of \( f(x) \) is \((-\infty, \infty)\).
  • Range: The range is the set of possible output values. For our cubic polynomial, as \( x \) gets large or goes negative, \( x^3 \) dominates, pushing the function's output towards both positive and negative infinity. Therefore, the range is also \((-\infty, \infty)\).
Grasping these concepts helps in understanding how functions behave and in solving various problems involving functions.
Increasing and Decreasing Intervals
Identifying where a function increases or decreases provides insights into its behavior over different intervals. We use the derivative \( f'(x) \) to determine this for the function \( f(x) = x^3 - 4x \).
Let's break it down:
  • Calculate the derivative: \( f'(x) = 3x^2 - 4 \).
  • Find critical points by setting the derivative to zero: \( 3x^2 - 4 = 0 \) gives \( x = \pm \sqrt{\frac{4}{3}} \), approximately \( x = \pm 1.15 \).
  • Test intervals around these critical points:
    • For \((-\infty, -1.15)\), choose a test point like \( x = -2 \). Here, \( f'(x) < 0 \), indicating the function is decreasing.
    • For \((-1.15, 1.15)\), a test point like \( x = 0 \) shows \( f'(x) > 0 \), meaning the function is increasing.
    • For \((1.15, \infty)\), test \( x = 2 \), where \( f'(x) < 0 \), indicating a decrease.
Hence, the function decreases from \(-\infty\) to \(-1.15\), increases from \(-1.15\) to \(1.15\), and decreases again from \(1.15\) to \(\infty\). Recognizing these intervals is key in understanding the shape and movement of polynomial functions in relation to their graphs.

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Most popular questions from this chapter

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