Examples of function in the following topics:
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- A power function is a function of the form f(x)=cxr where c and r are constant real numbers.
- A power function is a function of the form f(x)=cxr where c and r are constant real numbers.
- Polynomials are made of power functions.
- Functions of the form f(x)=x3, f(x)=x1.2, f(x)=x−4 are all power functions.
- Describe the relationship between the power functions and infinitely differentiable functions
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- Linear and quadratic functions make lines and a parabola, respectively, when graphed and are some of the simplest functional forms.
- They are one of the simplest functional forms.
- Linear functions may be confused with affine functions.
- However, the term "linear function" is quite often loosely used to include affine functions of the form f(x)=mx+b.
- A quadratic function, in mathematics, is a polynomial function of the form: f(x)=ax2+bx+c,a≠0.
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- An inverse function is a function that undoes another function: For a function f(x)=y the inverse function, if it exists, is given as g(y)=x.
- Inverse function is a function that undoes another function: If an input x into the function f produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., f(x)=y, and g(y)=x.
- If f is invertible, the function g is unique; in other words, there is exactly one function g satisfying this property (no more, no less).
- Not all functions have an inverse.
- A function f and its inverse f−1.
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- A continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output.
- A continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output.
- Otherwise, a function is said to be a "discontinuous function."
- A continuous function with a continuous inverse function is called "bicontinuous."
- This function is continuous.
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- A transcendental function is a function that is not algebraic.
- Examples of transcendental functions include the exponential function, the logarithm, and the trigonometric functions.
- A transcendental function is a function that "transcends" algebra in the sense that it cannot be expressed in terms of a finite sequence of the algebraic operations of addition, multiplication, power, and root extraction.
- Because of this, transcendental functions can be an easy-to-spot source of dimensional errors.
- Bottom panel: Graph of sine function versus angle.
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- An inverse function is a function that undoes another function.
- Stated otherwise, a function is invertible if and only if its inverse relation is a function on the range Y, in which case the inverse relation is the inverse function.
- Not all functions have an inverse.
- Let's take the function y=x2+2.
- We can check to see if this inverse "undoes" the original function by plugging that function in for x:
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- If every point of a function has a derivative, there is a derivative function sending the point a to the derivative of f at x=a: f′(a).
- This function is written f′(x) and is called the derivative function or the derivative of f.
- Using this idea, differentiation becomes a function of functions: The derivative is an operator whose domain is the set of all functions that have derivatives at every point of their domain and whose range is a set of functions.
- It is only defined on functions:
- At the point where the function makes a jump, the derivative of the function does not exist.
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- The derivative of the exponential function is equal to the value of the function.
- Functions of the form cex for constant c are the only functions with this property.
- The slope of the graph at any point is the height of the function at that point.
- The rate of increase of the function at x is equal to the value of the function at x.
- Graph of the exponential function illustrating that its derivative is equal to the value of the function.
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- The inverse trigonometric functions are also known as the "arc functions".
- There are three common notations for inverse trigonometric functions.
- The differentiation of trigonometric functions is the mathematical process of finding the rate at which a trigonometric function changes with respect to a variable.
- Thus each function has an infinite number of antiderivatives.
- Note that some of these functions are not valid for a range of x which would end up making the function undefined.
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- Also called vector functions, vector valued functions allow you to express the position of a point in multiple dimensions within a single function.
- A three-dimensional vector valued function requires three functions, one for each dimension.
- This is a three dimensional vector valued function.
- This can be broken down into three separate functions called component functions:
- This function is representing a position.