Examples of even function in the following topics:
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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.
- For example, the function $f(x)=x^2$ is even because it has an exponent, $2$, that is an even integer.
- How can we check if a function is odd or even?
- Remember the degree of the function, in this case a $4$ which is even, may not always dictate if the function is in fact even.
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- We have previously discussed even and odd functions.
- Recall that cosine is an even function because it is symmetric about the $y$-axis.
- Graphs that are symmetric about the $y$-axis represent even functions.
- For even functions, any two points with opposite $x$-values have the same function value.
- Cosine and secant are even functions, with symmetry around the $y$-axis.
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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:
- This indicates that it is an even function.
- For even functions, any two points with opposite $x$-values have the same function value.
- The cosine function is even, meaning it is symmetric about the
$y$-axis.
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- A rational function is any function which can be written as the ratio of two polynomial functions.
- Any function of one variable, $x$, is called a rational function if, and only if, it can be written in the form:
- Note that every polynomial function is a rational function with $Q(x) = 1$.
- A constant function such as $f(x) = \pi$ is a rational function since constants are polynomials.
- Note that the function itself is rational, even though the value of $f(x)$ is irrational for all $x$.
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- Below is a mapping of function $f(x)$ and its inverse function, $f^-1(x)$.
- In general, given a function, how do you find its inverse function?
- Since the function $f(x)=3x^2-1$ has multiple outputs, its inverse is actually NOT a function.
- Even though the blue (function) curve is a function (passes the vertical line test), its inverse (red) only includes the positive square root values and not the negative square root values of the function's range.
- A function's inverse may not always be a function.
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- In mathematics, a piecewise function
is a function in which more than one formula is used to define the output over different pieces of the domain.
- Piecewise functions are defined using the common functional notation, where the body of the function is an array of functions and associated intervals.
- After finding and plotting some ordered pairs for all parts ("pieces") of the function the result is the V-shaped curve of the absolute value function below.
- When $x=2$, the function is also piecewise continuous.
- Even though there looks like a gap from $y=1$ to $y=2$, the piece of the function $f(x)=x^2$ includes those values.
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- A root, or zero, of a polynomial function is a value that can be plugged into the function and yield $0$.
- The zero of a function, $f(x)$, refers to the value or values of $x$ that will result in the function equaling zero, $f(x)=0$.
- For example, if given the function:
- A real number is any rational or irrational number, such as $-5$, $\frac {4}{3}$, or even $\sqrt 2$.
- Even though all polynomials have roots, not all roots are real numbers.
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- From these, one can easily find critical values of the function by inspection.
- The method of graphing a function to determine general properties can be used to solve financial problems.Given the algebraic equation for a quadratic function, one can calculate any point on the function, including critical values like minimum/maximum and x- and y-intercepts.
- Suppose this models a profit function $f(x)$ in dollars that a company earns as a function of $x$ number of products of a given type that are sold, and is valid for values of $x$ greater than or equal to $0$ and less than or equal to $500$.
- If a financier wanted to find the number of sales required to break even, the maximum possible loss (and the number of sales required for this loss), and the maximum profit (and the number of sales required for this profit), they could simply reference a graph instead of calculating it out algebraically.
- The break-even points are between $15$ and $16$ sales, and between $484$ and $485$ sales.
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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.
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- If we take values of $x$ that are even closer to $0$, we can arrive at the following points: $(\frac{1}{b^{10}},-10),(\frac{1}{b^{100}},-100)$ and $(\frac{1}{b^{1000}},-1000)$
- The range of the function is all real numbers.
- At first glance, the graph of the logarithmic function can easily be mistaken for that of the square root function.
- When graphing without a calculator, we use the fact that the inverse of a logarithmic function is an exponential function.
- The graph of the logarithmic function with base $3$ can be generated using the function's inverse.