Analytic Geometry
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A function is a relation between a set of inputs and outputs where each input is mapped to exactly one output. The function notation y=f(x) represents an input x with its corresponding single output y value. Functions can be combined using operations like addition, subtraction, multiplication and division. Rational functions are fractions of polynomials. Exponential functions have the variable in the exponent. The trigonometric functions relate angles and ratios of the sides of a right triangle.
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This document discusses one-to-one functions and logarithmic functions. It defines one-to-one functions as those where each element of the domain corresponds to exactly one element in the range. It provides examples of functions that are and aren't one-to-one. The document also discusses inverse functions and how the inverse of an exponential function is a logarithmic function. It provides examples of evaluating logarithms and using properties of exponents and logarithms to simplify expressions.
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3) It discusses the properties of one-to-one, onto, and bijective functions and provides examples of each. The concepts of inverse and composition of functions are also introduced.
4) Useful mathematical functions like floor, ceiling, and round are defined. Concepts related to limits, continuity, differentiability of functions and their graphs are explained briefly.
5
12 x1 t05 01 inverse functions (2013)
The document discusses inverse functions. An inverse function f^-1(x) is obtained by interchanging x and y in the original function f(x). For f^-1(x) to be a function, there must be a unique y-value for each x-value. A function and its inverse are reflections across the line y=x. The domain of f(x) is the range of f^-1(x), and vice versa. To test if an inverse function exists, use the horizontal line test or check if rewriting the inverse relation as y=g(x) yields a unique expression for y. If an inverse function exists, f^-1(f(x)) = x and f(
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The document discusses inverse functions. An inverse function f^-1(x) is obtained by interchanging x and y in the original function f(x). For f^-1(x) to be a function, there must be a unique y-value for each x-value. A function and its inverse are reflections across the line y=x. The domain of f(x) is the range of f^-1(x), and vice versa. Inverse functions satisfy the properties f^-1(f(x)) = x and f(f^-1(x)) = x. Examples are provided to demonstrate testing if an inverse function exists.
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The document discusses inverse functions. An inverse function f^-1(x) is obtained by interchanging x and y in the original function f(x). For f^-1(x) to be a function, there must be a unique y-value for each x-value. A function and its inverse are reflections across the line y=x. The domain of f(x) is the range of f^-1(x), and vice versa. To test if an inverse function exists, use the horizontal line test or check if rewriting the inverse relation as y=g(x) yields a unique expression for y. If an inverse function exists, f^-1(f(x)) = x and f(
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This document discusses inverse functions and their derivatives. It defines inverse functions as switching the x- and y-values of a function to "undo" the original function. A function has an inverse only if it passes the horizontal line test. The derivative of an inverse function at a point equals the reciprocal of the derivative of the original function at the corresponding point.
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* Find the value of a function.
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* Use the vertical line test to identify functions.
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- 2. What is a function? It is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. With the function notation y = f(x), each x value has only one corresponding y value. The x-values are the inputs, and the y-values are the outputs. 1 2 3 4 5 4 3 2
- 3. Sum of f and g: (f + g)(x) = f(x) + g(x) Difference of f and g: (f - g)(x) = f(x) - g(x) Product of f and g: (f . g)(x) = f(x) . g(x) Quotient of f and g: (f/g)(x) = f(x)/g(x), g(x) not equal to 0
- 13. These functions are the ratio of two polynomials. One field of study where they are important is in stability analysis of mechanical and electrical systems (which uses Laplace transforms).
- 14. A rational function is a fraction of polynomials. That is, if p(x) and q(x) are polynomials, thenp(x) q(x)
- 16. A function of the form f(x) = abx where a = 0 and b>0 ; b = 1 are real numbers. Exponential functions are functions where the variable is in the exponent.
- 17. bx is the inverse function of logb(x)
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