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Function

Learn about functions, one-to-one properties, and inverse functions through mathematical concepts, including the horizontal line test and derivative test.

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Function

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  1. Function A functionf is a rule that assigns to each element x in a set D exactly one element, called f(x), in a set C. D: The domain of f is the set D of elements on which f is defined. R: The range of f is the set of all possible values f(x) when x ranges over all possible values in the domain D. C: The codomain of f is the set C used to hold all possible values of f(x). We say that fmapsx to f(x), or that f(x) is the image of x under f.

  2. One-to-one function A function f is called one-to-one (1-to-1, injective) if it never takes on the same value twice, i.e., f(x1)≠ f(x2) whenever x1≠ x2. “Two-to-Two”: f always maps two distinct values to two distinct images Horizontal Line Test Assume a function f can be graphed on the Cartesian coordinate axes. The function f is one-to-one if and only if no horizontal line intersects its graph more than once.

  3. Three tests for one-to-one • For a function f, there are three tests to see if f is one-to-one • Horizontal line test. Pass <=> f is one-to-one • Algebraic test: • Assume x1≠ x2. Use algebra to show f(x1)≠f(x2). Conclude that f is one-to-one. • (Equivalent method)Assume f(x1)=f(x2). Use algebra to show x1≠ x2. Conclude that f is one-to-one. • Derivative test. If f has a derivative on its entire domain, take the derivative. • If the derivative is always positive, conclude f is one-to-one. • If the derivative is always negative, conclude f is one-to-one. • If the derivative is sometimes positive and sometimes negative and there are no discontinuities in the function, then the function is not one-to-one. • Warning: Passing the derivative test implies one-to-one.Failing the derivative test tells you nothing (inconclusive).

  4. Inverse of a function Let f be a one-to-one function with domain A and range B. Then its inverse function f-1 has domain B and range A and is defined by f-1(y)=x if and only if f(x)=y for any y in B. Domain of f-1 = range of f Range of f-1 = domain of f f-1(x)=y if and only if f(y)=x f-1(f(x))=x for every x in A f(f-1(x))=x for every x in B

  5. Derivative of the inverse of f Let g be the inverse of f. The tangent line L to f at P has reciprocal slope to the tangent line L’ to g at P’. i.e., slope(L) = 1/slope(L’) f’(b) = h/w=tan ө g’(a)= w/h=tan φ= cot ө So, g’(a)=1/tan ө=1/f’(b) = 1/f’(g(a)) L’ w ө φ L b h w h ө a a b (original image from ActiveMath.org)

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