1 precalculus i dr. claude s. moore danville community college composite and inverse functions...
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PRECALCULUS I
Dr. Claude S. MooreDanville Community
College
Composite and Inverse Functions•Translation, combination, composite
•Inverse, vertical/horizontal line test
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)
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)
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
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)
Product: (f g) = f(x)g(x)
4. Quotient:
Arithmetic Combinations
0)(,)(
)()(
xg
xg
xfx
g
f
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
(fg)(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
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.
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.
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.
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
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 gand in the domain of f.
Definition of an Inverse Function
If the function f has an inverse function g, then
domain range
f x yg x y
Relationship between Domains and Ranges of f and g
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