1

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