odd function

(noun)

A continuous set of (x,f(x))\left(x,f(x)\right)(x,f(x)) points for which f(−x)=−f(x)f(-x) = -f(x)f(−x)=−f(x), and there is symmetry about the origin.

Related Terms

  • Periodic Functions
  • vertical asymptotes
  • periodic function
  • even function
  • period
  • vertical asymptote
  • asymptote

(noun)

An continuous set of (x,f(x))\left(x, f(x)\right)(x,f(x)) points in which f(−x)=−f(x)f(-x) = -f(x)f(−x)=−f(x), and there is symmetry about the origin.

Related Terms

  • Periodic Functions
  • vertical asymptotes
  • periodic function
  • even function
  • period
  • vertical asymptote
  • asymptote

(noun)

A continuous set of (x,f(x))\left(x, f(x)\right)(x,f(x)) points in which f(−x)=−f(x)f(-x) = -f(x)f(−x)=−f(x), with symmetry about the origin.

Related Terms

  • Periodic Functions
  • vertical asymptotes
  • periodic function
  • even function
  • period
  • vertical asymptote
  • asymptote

Examples of odd function in the following topics:

  • Trigonometric Symmetry Identities

    • We have previously discussed even and odd functions.
    • Recall that even functions are symmetric about the yyy-axis, and odd functions are symmetric about the origin, (0,0)(0, 0)(0,0).
    • On the other hand, sine and tangent are odd functions because they are symmetric about the origin.
    • Graphs that are symmetric about the origin represent odd functions.
    • For odd functions, any two points with opposite xxx-values also have opposite yyy-values.
  • Even and Odd Functions

    • Functions can be classified as "odd" or "even" based on their composition.
    • The terms "odd" and "even" can only be applied to a limited set of functions.
    • Oftentimes, the parity of a function will reveal whether it is odd or even.
    • How can we check if a function is odd or even?  
    • The function, f(x)=x3−4xf(x)=x^3-4xf(x)=x​3​​−4x is odd since the graph is symmetric about the origin.  
  • Sine and Cosine as Functions

    • A periodic function is a function with a repeated set of values at regular intervals.
    • Specifically, it is a function for which a specific horizontal shift, PPP, results in a function equal to the original function:
    • As we can see in the graph of the sine function, it is symmetric about the origin, which indicates that it is an odd function.
    • This is characteristic of an odd function: two inputs that are opposites have outputs that are also opposites.
    • The sine function is odd, meaning it is symmetric about the origin.
  • Tangent as a Function

    • Characteristics of the tangent function can be observed in its graph.
    • The tangent function can be graphed by plotting (x,f(x))\left(x,f(x)\right)(x,f(x)) points.
    • As with the sine and cosine functions, tangent is a periodic function.
    • In the graph of the tangent function on the interval −π2\displaystyle{-\frac{\pi}{2}}−​2​​π​​ to π2\displaystyle{\frac{\pi}{2}}​2​​π​​, we can see the behavior of the graph over one complete cycle of the function.
    • The graph of the tangent function is symmetric around the origin, and thus is an odd function.
  • The Leading-Term Test

    • Consider the polynomial function:
    • If nnn is odd and ana_na​n​​ is positive, the function declines to the left and inclines to the right.
    • If nnn is odd and ana_na​n​​ is negative, the function inclines to the left and declines to the right.
    • Except when xxx is negative and nnn is odd; then the opposite is true.
    • Because the degree is odd and the leading coefficient  is positive, the function declines to the left and inclines to the right.
  • Other Equations in Quadratic Form

    • Many equations with no odd-degree terms can be reduced to quadratics and solved with the same methods as quadratics.
    • For example, if a quartic equation is biquadratic—that is, it includes no terms of an odd-degree— there is a quick way to find the zeroes of the quartic function by reducing it into a quadratic form.  
    • Consider a quadratic function with no odd-degree terms which has the form:
  • Finding Zeros of Factored Polynomials

    • A polynomial function may have many, one, or no zeros.
    • All polynomial functions of positive, odd order have at least one zero (this follows from the fundamental theorem of algebra), while polynomial functions of positive, even order may not have a zero (for example x4+1x^4+1x​4​​+1 has no real zero, although it does have complex ones).
    • Regardless of odd or even, any polynomial of positive order can have a maximum number of zeros equal to its order.
    • For example, a cubic function can have as many as three zeros, but no more.
    • Thus if you have found such a factorization of a given function, you can be completely sure what the zeros of that function are.
  • Basics of Graphing Polynomial Functions

    • A polynomial function in one real variable can be represented by a graph.
    • A typical graph of a polynomial function of degree 3 is the following:
    • Functions of odd degree will go to negative or positive infinity when xxx goes to negative infinity and vice versa, again depending on the highest-degree term coefficient.
    • Every time we cross a zero of odd multiplicity (if the number of zeros equals the degree of the polynomial, all zeros have multiplicity one and thus odd multiplicity) we change sign.
    • We also call this the yyy-intercept of the function.
  • The Rule of Signs

    • The difference is that you must start by finding the coefficients of odd power (for example, x3x^3x​3​​ or x5x^5x​5​​, but not x2x^2x​2​​ or x4x^4x​4​​).
    • This can also be done by taking the function, f(x)f(x)f(x), and substituting the xxx for −x-x−x, so that we have the function f(−x)f(-x)f(−x).
    • By only multiplying the odd powered coefficients by −1-1−1, we are essentially saving ourselves a step.
    • This function has one sign change between the second and third terms.
    • Change the exponents of the odd-powered coefficients, remembering to change the sign of the first term.
  • Basics of Graphing Exponential Functions

    • The exponential function y=bxy=b^xy=b​x​​ where b>0b>0b>0 is a function that will remain proportional to its original value when it grows or decays.
    • At the most basic level, an exponential function is a function in which the variable appears in the exponent.
    • The most basic exponential function is a function of the form y=bxy=b^xy=b​x​​ where bbb is a positive number.
    • If b=1b=1b=1, then the function becomes y=1xy=1^xy=1​x​​.
    • If bbb is negative, then raising bbb to an even power results in a positive value for yyy while raising bbb to an odd power results in a negative value for yyy, making it impossible to join the points obtained an any meaningful way and certainly not in a way that generates a curve as those in the examples above.
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