Inverse composite functions


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Inverse composite functions

  1. 1. 1 PRECALCULUS I Dr. Claude S. Moore Danville Community College Composite and Inverse Functions •Translation, combination, composite •Inverse, vertical/horizontal line test
  2. 2. For a positive real number c, vertical shifts of y = f(x) are: 1. Vertical shift c units upward: h(x) = y + c = f(x) + c 2. Vertical shift c units downward: h(x) = y − c = f(x) − c Vertical Shifts (rigid transformation)
  3. 3. For a positive real number c, horizontal shifts of y = f(x) are: 1. Horizontal shift c units to right: h(x) = f(x − c) ; x − c = 0, x = c 2. Vertical shift c units to left: h(x) = f(x + c) ; x + c = 0, x = -c Horizontal Shifts (rigid transformation)
  4. 4. Reflections in the coordinate axes of the graph of y = f(x) are represented as follows. 1. Reflection in the x-axis: h(x) = −f(x) (symmetric to x-axis) 2. Reflection in the y-axis: h(x) = f(−x) (symmetric to y-axis) Reflections in the Axes
  5. 5. Let x be in the common domain of f and g. 1. Sum: (f + g)(x) = f(x) + g(x) 2. Difference: (f − g)(x) = f(x) − g(x) 3. Product: (f ⋅ g) = f(x)⋅g(x) 4. Quotient: Arithmetic Combinations 0)(, )( )( )( ≠=      xg xg xf x g f
  6. 6. The domain of the composite function f(g(x)) is the set of all x in the domain of g such that g(x) is in the domain of f. The composition of the function f with the function g is defined by (fg)(x) = f(g(x)). Two step process to find y = f(g(x)): 1. Find h = g(x). 2. Find y = f(h) = f(g(x)) Composite Functions
  7. 7. One-to-One Function For y = f(x) to be a 1-1 function, each x corresponds to exactly one y, and each y corresponds to exactly one x. A 1-1 function f passes both the vertical and horizontal line tests.
  8. 8. VERTICAL LINE TEST for a Function A set of points in a coordinate plane is the graph of y as a function of x if and only if no vertical line intersects the graph at more than one point.
  9. 9. HORIZONTAL LINE TEST for a 1-1 Function The function y = f(x) is a one-to-one (1-1) function if no horizontal line intersects the graph of f at more than one point.
  10. 10. A function, f, has an inverse function, g, if and only if (iff) the function f is a one-to-one (1-1) function. Existence of an Inverse Function
  11. 11. A function, f, has an inverse function, g, if and only if f(g(x)) = x and g(f(x)) = x, for every x in domain of g and in the domain of f. Definition of an Inverse Function
  12. 12. If the function f has an inverse function g, then domain range f x y g x y Relationship between Domains and Ranges of f and g
  13. 13. 1. Given the function y = f(x). 2. Interchange x and y. 3. Solve the result of Step 2 for y = g(x). 4. If y = g(x) is a function, then g(x) = f-1 (x). Finding the Inverse of a Function