Examples of injective function in the following topics:
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- A one-to-one function, also called an injective function, never maps distinct elements of its domain to the same element of its codomain.
- A one-to-one function, also called an injective function, never maps distinct elements of its domain to the same element of its co-domain.
- Occasionally, an injective function from $X$ to $Y$ is denoted $f: X \mapsto Y$, using an arrow with a barbed tail.
- One way to check if the function is one-to-one is to graph the function and perform the horizontal line test.
- The graph of the function $f(x)=x^2$ fails the horizontal line test and is therefore NOT a one-to-one function.
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- A rational function is one such that $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$; the domain of a rational function can be calculated.
- 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.
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- Each trigonometric function has an inverse function that can be graphed.
- In order to use inverse trigonometric functions, we need to understand
that an inverse trigonometric function “undoes” what the original
trigonometric function “does,” as is the case with any other function
and its inverse.
- Note that the domain of the inverse function is
the range of the original function, and vice versa.
- However, the sine, cosine, and tangent functions are not
one-to-one functions.
- As with other
functions that are not one-to-one, we will need to restrict the domain of each function to yield a new function that is one-to-one.
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- As part of exploring how functions change, we can identify intervals over which the function is changing in specific ways.
- We say that a function is increasing on an interval if the function values increase as the input values increase within that interval.
- In terms of a linear function $f(x)=mx+b$, if $m$ is positive, the function is increasing, if $m$ is negative, it is decreasing, and if $m$ is zero, the function is a constant function.
- In mathematics, a constant function is a function whose values do not vary, regardless of the input into the function.
- A function is a constant function if $f(x)=c$ for all values of $x$ and some constant $c$.
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- A translation of a function is a shift in one or more directions.
- To translate a function vertically is to shift the function up or down.
- The original function we will use is:
- To translate a function horizontally is the shift the function left or right.
- Again, the original function is:
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- Functional composition allows for the application of one function to another; this step can be undone by using functional decomposition.
- The process of combining functions so that the output of one function becomes the input of another is known as a composition of functions.
- The resulting function is known as a composite function.
- Note that the range of the inside function (the first function to be evaluated) needs to be within the domain of the outside function.
- Practice function composition by applying the rules of one function to the results of another function
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- Function notation, $f(x)$ is read as "$f$ of $x$" which means "the value of the function at $x$."
- This function is that of a line, since the highest exponent in the function is a $1$, so simply connect the three points.
- The graph for this function is below.
- The degree of the function is 3, therefore it is a cubic function and is sometimes shaped like the letter N.
- The function is linear, since the highest degree in the function is a $1$.
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- A composite function represents, in one function, the results of an entire chain of dependent functions.
- In mathematics, function composition is the application of one function to the results of another.
- In general, composition of functions will not be commutative.
- A composite function
represents in one function the results of an entire chain of dependent functions.
- If $f$ is an invertible function with domain $X$ and range $Y$, then
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- Recognize whether a function has an inverse by using the horizontal line test