Question

Multiple Choice The cube function \(f(x)=x^{3}\) is _____. (a) even (b) odd (c) neither The graph of the cube function _____. (a) has no symmetry (b) is symmetric about the \(y\) -axis (c) is symmetric about the origin (d) is symmetric about the line \(y=x\)

Short answer

The cube function is odd and its graph is symmetric about the origin.

Step-by-step solution

Define Even and Odd Functions

A function is even if it satisfies the condition: \[ f(-x) = f(x) \]A function is odd if it satisfies the condition: \[ f(-x) = -f(x) \]

Apply the Definitions to the Given Function

The given function is \[ f(x) = x^3 \]First, evaluate \[ f(-x) = (-x)^3 = -x^3 \]Thus, \[ f(-x) = -f(x) \]This shows that the function is odd.

Determine the Symmetry of the Graph

Since the function \[ f(x) = x^3 \]is odd, we know that its graph is symmetric about the origin. This means that if a point \[ (a, b) \] is on the graph, then \[ (-a, -b) \] is also on the graph.

Key concepts

Odd Functions

Understanding odd functions is essential when analyzing symmetry in mathematical functions. An odd function is one where the function satisfies the condition
\[ f(-x) = -f(x) \]
This shows that for any input value \(x\), if you input its negative \(-x\), the output is the negative of the original function value.

Consider the cube function \(f(x) = x^3\). If we input \(-x\) into this function, we get \[ f(-x) = (-x)^3 = -x^3 \]
Notice that\( f(-x) = -f(x) \). This confirms that the cube function is indeed an odd function. Recognizing this property helps identify the function's symmetry.

Even Functions

Even functions are another important type of mathematical function that show a different type of symmetry. A function is considered even if it meets the condition:
\[ f(-x) = f(x) \]
This means that for every input value \(x\) and its negative \(-x\), the function's output remains the same.

A typical example of an even function is \(f(x) = x^2\). Testing the function, we find: \[ f(-x) = (-x)^2 = x^2 \]
Since \(f(-x) = f(x)\), \(x^2\) is an even function. Unlike odd functions, even functions have symmetry about the y-axis, as we'll see in the next section.

Function Symmetry

Symmetry in functions is a fascinating topic. It allows us to visualize how functions behave across different axes.
Odd functions, like \(f(x) = x^3\), are symmetric about the origin. What does this mean? If a point \((a, b)\) lies on the graph of an odd function, then the point \((-a,-b)\) will also lie on the graph. This type of symmetry is important in higher mathematics because it can simplify complex equations and problems.

On the other hand, even functions show symmetry about the y-axis. This means for any point \((a, b)\) on the graph, the point \((-a, b)\) will also be on the graph. This is often easier to visualize and has practical applications in physics and engineering.

Understanding these properties can greatly enhance your problem-solving skills, particularly when dealing with different types of functions in calculus and algebra.

Graph Symmetry

Graphing functions helps to visually understand their properties and symmetries. Let's talk about graph symmetry specifically.
For odd functions like the cube function \(f(x) = x^3\), the graph will be symmetric about the origin. This implies that rotating the graph 180 degrees around the origin will make it look unchanged. For instance, if you have a point \((2, 8)\) on the graph, you'll also have the point \((-2, -8)\) on the graph. This is a key feature of odd functions.
For even functions, the graph will be symmetric about the y-axis. So, flipping the graph over the y-axis will result in an identical graph. If you consider the function \(f(x) = x^2\), you can see that the points \((2, 4)\) and \((-2, 4)\) lie on the graph, clearly showing y-axis symmetry.
Recognizing these symmetries helps in quickly sketching the graphs of functions and understanding their behavior, which is particularly useful in calculus and advanced mathematics.