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