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Chapter 3: Problem 61

61鈥�68 ? Determine whether the function f is even, odd, or neither. If f is even or odd, use symmetry to sketch its graph. $$f(x) = x^2$$

Short Answer

Expert verified
The function \(f(x) = x^2\) is even and symmetric around the y-axis.

Step by step solution

01

Understanding Even Functions

A function is called even if, for every value of \(x\) in the domain of the function, \(f(-x) = f(x)\). Even functions have symmetry around the y-axis.
02

Calculate f(-x)

For the function \(f(x) = x^2\), substitute \(-x\) in place of \(x\) to find \(f(-x)\). We have \(f(-x) = (-x)^2\). Since squaring a negative number results in a positive number, \((-x)^2 = x^2\).
03

Comparing f(x) and f(-x)

With \(f(x) = x^2\) and \(f(-x) = x^2\), it is evident that \(f(-x) = f(x)\) for all \(x\). This satisfies the condition for even functions.
04

Conclusion About Symmetry

Since \(f(x) = x^2\) meets the condition for an even function, it is symmetric around the y-axis. Graphically, when you plot the function, for every point \((x, f(x))\), there exists a corresponding point \((-x, f(x))\) making it symmetric.

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Key Concepts

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

Function Symmetry
Symmetry in functions is a fascinating topic because it helps us understand how a graph behaves across certain axes. When dealing with functions, symmetry plays a key role in simplifying their analysis.

There are different types of symmetry:
  • Y-Axis Symmetry: If a function is described as even, it has y-axis symmetry. This means that if you "fold" the graph along the y-axis, both sides will match perfectly.
  • Origin Symmetry: Associated with odd functions, this symmetry means rotating the graph 180 degrees around the origin will result in the same graph.
To determine symmetry, you can check:
  • If a function is even, then for all values of x, the equation \(f(-x) = f(x)\) holds true.
  • If a function is odd, you will find that \(f(-x) = -f(x)\).
Recognizing these patterns can make sketching graphs much easier. Symmetrical graphs are neat and easier to interpret at a glance.
Parabola Graph
When you think of a parabola, imagine the smooth, U-shaped curve that is a hallmark of quadratic functions, such as our function \(f(x) = x^2\). Parabolas are the graphs of quadratic equations, and they have some distinct characteristics:

  • Vertex: The turning point of the parabola. In the case of \(f(x) = x^2\), this is at the origin (0,0).
  • Axis of Symmetry: Parabolas have a vertical line running through the vertex which divides the graph into two mirror images. For \(f(x) = x^2\), this is the line \(x = 0\), or the y-axis.
  • Opening: Parabolas can open upwards or downwards. In an equation like \(f(x) = x^2\), the curve opens upwards.
The symmetry of parabolas makes them easy to sketch if we know a few key points, such as the vertex and the direction of opening. Observing a parabola, you'll also notice that every change in \(x\) is reflected evenly across the axis of symmetry.
Even and Odd Functions
Functions are classified into two categories based on their symmetry properties: even and odd. Understanding their characteristics helps you determine the shape and balance of their graphs.

  • Even Functions: As previously mentioned, even functions satisfy the condition \(f(-x) = f(x)\). Their graphs are symmetric about the y-axis, which means that they provide a mirrored view on either side of this axis.
  • Odd Functions: If a function fulfills \(f(-x) = -f(x)\), it's classified as an odd function. The graph of an odd function will look the same after a 180-degree rotation around the origin, showcasing symmetry about the origin.
  • Neither: Some functions do not meet the criteria for being even or odd, so they lack those specific types of symmetry.
Knowing if a function is even or odd not only helps in sketching but also in understanding the function's behavior at various points. In our example, \(f(x) = x^2\) is a quintessential even function, which is why its graph is perfectly symmetric about the y-axis.

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