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Chapter 2: Problem 87

Use the analyric method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use \(a\) calculator and the standand window to support your conclusion. $$f(x)=\frac{1}{4 x^{3}}$$

Short Answer

Expert verified
The function is symmetric about the origin.

Step by step solution

01

Test for Symmetry About the y-axis

To check if the function is symmetric about the y-axis, replace \(x\) with \(-x\) and see if the resulting expression is the same as \(f(x)\). Calculate \(f(-x)\): \[ f(-x) = \frac{1}{4(-x)^3} = \frac{1}{-4x^3} = -\frac{1}{4x^3} \]Since \(f(-x) eq f(x)\), the function is not symmetric about the y-axis.
02

Test for Symmetry About the Origin

To check if the function is symmetric about the origin, replace \(x\) with \(-x\) and check if \(f(-x) = -f(x)\). We have already found \(f(-x) = -\frac{1}{4x^3}\). Now, calculate \(-f(x)\):\[-f(x) = -\left(\frac{1}{4x^3}\right) = -\frac{1}{4x^3} \]Since \(f(-x) = -f(x)\), the function is symmetric about the origin.
03

Graph the Function to Verify Symmetry

Plot the function \(f(x) = \frac{1}{4x^3}\) using a calculator. Observe the graph: It should appear the same on opposite quadrants, confirming that it is symmetric about the origin. There are no reflections over the y-axis.

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Key Concepts

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

Graphing Functions
Graphing a function involves plotting its points on a coordinate system, represented usually by the x-axis and y-axis. The graph of a function provides a visual representation of the behavior of the function. Here are some general steps to graphing a function:
  • Choose several values of x, both positive and negative, to substitute into the function.
  • Calculate the corresponding y values.
  • Plot the points on a Cartesian plane.
  • Connect the points smoothly, following the trend indicated by the values.
When working with more complex functions, like rational or polynomial functions, using a graphing calculator or software can be extremely helpful to get an accurate picture. In the case of the function \(f(x) = \frac{1}{4x^3}\), understanding its graph helps in analyzing its symmetry properties.
Symmetry About the Origin
A function is symmetric about the origin if, for every point \((x, y)\) on the graph, the point \((-x, -y)\) is also on the graph. Mathematically, this means that \(f(-x) = -f(x)\) for every x in the domain of the function.
To determine if a graph is symmetric about the origin, follow these steps:
  • Calculate \(f(-x)\) by replacing x with -x in the function.
  • Compare \(f(-x)\) with \(-f(x)\).
For the function \(f(x) = \frac{1}{4x^3}\), when \(x\) is replaced with \(-x\), the result \(f(-x) = -\frac{1}{4x^3}\) matches \(-f(x)\), confirming its origin symmetry. This means the graph will look exactly the same if rotated 180 degrees around the origin.
Symmetry About the y-axis
Symmetry about the y-axis means that a function's graph reflects itself across the y-axis. For a function to have this symmetry property, the condition \(f(-x) = f(x)\) must hold true for every x in its domain.
Here鈥檚 how to test for y-axis symmetry:
  • Replace x with -x in the function to find \(f(-x)\).
  • Check if \(f(-x)\) is equal to \(f(x)\).
In the example of the function \(f(x) = \frac{1}{4x^3}\), substituting \(-x\) results in \(f(-x) = -\frac{1}{4x^3}\), not equal to \(f(x)\). Hence, this function does not exhibit symmetry about the y-axis. It's a useful test to quickly determine the nature of symmetry in polynomial and rational functions.

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Most popular questions from this chapter

For certain pairs of functions \(f\) and \(g .(f \circ g)(x)=x\) and \((g \circ f)(x)=x\). Show that this is true for the pairs in Exercises \(65-68\). $$f(x)=4 x+2, g(x)=\frac{1}{4}(x-2)$$

If \(f(x)=k, k \neq 0\) is a constant function and \(g(x)=m x+b, m \neq 0\) is a linear function, then determine the range of the composition \((f \circ g)(x)\)

Determine the difference quotient \(\frac{f(x+h)-f(x)}{h}\) (where \(h \neq 0\) j for each function \(f\). Simplify completely. $$f(x)=\frac{1}{2 x}$$

For certain pairs of functions \(f\) and \(g .(f \circ g)(x)=x\) and \((g \circ f)(x)=x\). Show that this is true for the pairs in Exercises \(65-68\). $$f(x)=\sqrt[3]{5 x+4}, g(x)=\frac{1}{5} x^{3}-\frac{4}{5}$$

Solve each equation or inequality graphically. $$2 x+8>-|3 x+4|$$
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