Examples of odd function in the following topics:
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- We have previously discussed even and odd functions.
- Recall that even functions are symmetric about the $y$-axis, and odd functions are symmetric about the origin, $(0, 0)$.
- 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 $x$-values also have opposite $y$-values.
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- 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, $f(x)=x^3-4x$ is odd since the graph is symmetric about the origin.
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- 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, $P$, 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.
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- Characteristics of the tangent function can be observed in its graph.
- The tangent function can be graphed by plotting $\left(x,f(x)\right)$ points.
- As with the sine and cosine functions, tangent is a periodic function.
- In the graph of the tangent function on the interval $\displaystyle{-\frac{\pi}{2}}$ to $\displaystyle{\frac{\pi}{2}}$, 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.
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- Consider the polynomial function:
- If $n$ is odd and $a_n$ is positive, the function declines to the left and inclines to the right.
- If $n$ is odd and $a_n$ is negative, the function inclines to the left and declines to the right.
- Except when $x$ is negative and $n$ 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.
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- 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:
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- 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 $x^4+1$ 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.
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- 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 $x$ 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 $y$-intercept of the function.
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- The difference is that you must start by finding the coefficients of odd power (for example, $x^3$ or $x^5$, but not $x^2$ or $x^4$).
- This can also be done by taking the function, $f(x)$, and substituting the $x$ for $-x$, so that we have the function $f(-x)$.
- By only multiplying the odd powered coefficients by $-1$, 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.
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- The exponential function $y=b^x$ where $b>0$ 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 $y=b^x$ where $b$ is a positive number.
- If $b=1$, then the function becomes $y=1^x$.
- If $b$ is negative, then raising $b$ to an even power results in a positive value for $y$ while raising $b$ to an odd power results in a negative value for $y$, 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.