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PRECALCULUS I

PRECALCULUS I. Composite and Inverse Functions Translation, combination, composite Inverse, vertical/horizontal line test. Dr. Claude S. Moore Danville Community College. Vertical Shifts (rigid transformation). For a positive real number c , vertical shifts of y = f(x) are:

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PRECALCULUS I

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  1. PRECALCULUS I • Composite and Inverse Functions • Translation, combination, composite • Inverse, vertical/horizontal line test Dr. Claude S. MooreDanville Community College

  2. Vertical Shifts(rigid transformation) For a positive real number c, vertical shifts of y = f(x) are: 1. Vertical shift c unitsupward: h(x) = y + c = f(x) + c 2. Vertical shift c units downward: h(x) = y - c = f(x) - c

  3. Horizontal Shifts(rigid transformation) For a positive real number c, horizontal shifts of y = f(x) are: 1. Horizontal shift c unitsto 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

  4. Reflections in the Axes 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)

  5. Arithmetic Combinations 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:

  6. Composite Functions 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 (fug)(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))

  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. 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. 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. Existence of an Inverse Function A function, f, has an inverse function, g, if and only if (iff) the function f is a one-to-one (1-1) function.

  11. Definition of an Inverse Function A function, f, has an inverse function, g, if and only iff(g(x)) = x and g(f(x)) = x,for every x in domain of gand in the domain of f.

  12. Relationship between Domains and Ranges of f and g If the function f has an inverse function g, then domain range f x y g x y

  13. Finding the Inverse of a Function 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).

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