Operations on Functions
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Functions, Parabolas, and Circles
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Peter Lo
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M014 © Peter Lo 2002
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Composite Functions
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Given two functions f and g, the Composite Function ,
denoted by f o g is defined by
(f o g)(x) = f [g(x)]
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The domain of f o g is the set of all numbers x in the
domain of g such that g(x) is in the domain of f.
M014 © Peter Lo 2002
The Sum of f and g, f + g is defined by
(f + g)(x) = f (x) + g(x)
The Difference of f and g, f – g is defined by
(f – g)(x) = f (x) – g(x)
The Product of f and g, f g is defined by
(f g)(x) = f (x) • g(x)
The Sum of f and g, f / g is defined by
(f / g)(x) = f (x) / g(x)
M014 © Peter Lo 2002
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Example
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Given f (x) = 2x2 – 3 and g(x) = 4x. Find:
u (f o g)(1)
u (g o f )(1)
u (f o f )(-2)
u (g o g)(-1)
M014 © Peter Lo 2002
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One-to-One Function
Inverse Function
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Let f: A→B be a function. The function f is called
an Injective Function, or an Injection if ∀x, y∈A,
f (x) = f (y) ⇒ x = y.
An injective function is also called a One-to-one
or 1-1 Function.
M014 © Peter Lo 2002
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The inverse of a one-to-one function f is the
function f -1 , which is obtained from f by
interchanging the coordinates in each ordered pair
of f.
M014 © Peter Lo 2002
Horizontal-Line Test
Identifying Inverse Functions
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A function is invertible if and only if no horizontal
line crosses its graph more than one.
M014 © Peter Lo 2002
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Function f and g are inverses of each other if and
only if
u (g ° f ) (x) = x for every number x in the
domain of f;
u (f ° g) (x) = x for every number x in the domain
of g.
M014 © Peter Lo 2002
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Switch-and-Solve Strategy for
Finding f -1
Example
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Replace f (x) by y.
Interchange x and y.
Express the equation for y.
Replace y by f -1 (x)
M014 © Peter Lo 2002
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Example
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Find the inverse of the function f (x) = x + 3.
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Direct Variation
The function f (x) = x3 – 3 is one-to-one. Find its inverse.
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Inverse Variation
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Joint Variation
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More Variation
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Example
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Supposes that y varies directly as x and the square
root of w and inversely as the square of v. If y = 2
when x = 2, w = 9 and v = 1. Find y as a function
of x, w and v.
M014 © Peter Lo 2002
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Conic Sections
Parabola
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Characteristic of y = (x - h)2 + k
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Characteristic of y = ax2 + bx + c
The graph of the equation y = (x - h) 2 + k (for a ≠ 0) is a
parabola with vertex (h, k), focus (h, k + p), and directrix y
= k – p, where a = 1/4p. If a > 0, the parabola open upward;
if a < 0, the parabola opens downward.
M014 © Peter Lo 2002
Given a line (the Directrix) and a point not on the
line (the Focus), the set of all points in the plane
that are equidistant from the point and the line is
called a Parabola.
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The graph of y = ax2 + bx + c (for a ≠ 0) is a parabola
opening upward if a > o and downward if a < 0. The xcoordinate of the vertex is x = -b/2a.
M014 © Peter Lo 2002
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Graphing a Parabola
Example
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Graph the parabola y = 3x2 + 6x + 1
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Circle
Standard Equation for a Circle
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A Circle is the set of all points in a plane that lie a
fixed distance from a given point in the plane. The
fixed distance is called the radius, and the given
points is called the Center.
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The graph of the equation
u (x – h)2 + (y – k)2 = r2
with r > 0, is a circle with center (h, k) and radius r.
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Example
Intersection of a line and a circle
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Find the center and radius of the circle given by
2x2 – 3x + 2y 2 + 7y = –5
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An Ellipse is the set of all points in a plane such that the
sum of their distances from two fixed points is a constant.
Each fixed point is call a Focus.
M014 © Peter Lo 2002
M014 © Peter Lo 2002
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Equation of an Ellipse centered at
Origin
Ellipse
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Find the intersection of the following equation:
u (x – 3) 2 + (y + 1)2 = 9
u y = x -1
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An Ellipse centered at (0, 0) with focus at (c, 0)
and constant sum 2a has equation:
x2 y2
+
=1
a 2 b2
where a, b and c are positive real numbers with
c2 = a2 – b 2 .
M014 © Peter Lo 2002
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Equation of an Ellipse centered at
(h, k)
Example
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An ellipse centered at (h, k) has equation
( x − h )2 ( y − k ) 2
+
=1
a2
b2
where a and b are positive real numbers.
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Example
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Hyperbola
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A hyperbola is the set of all points in the plane
such that the difference at their distances from two
fixed points (focus) is constant.
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Equation of a Hyperbola centered
at Origin
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Graphing Hyperbola
An Hyperbola centered at (0, 0) with focus at (c, 0)
and (-c, 0) and constant difference 2a has equation:
x2 y2
−
=1
a2 b2
where a, b and c are positive real numbers with
c2 = a2 + b 2 .
M014 © Peter Lo 2002
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Graphing Hyperbola
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M014 © Peter Lo 2002
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Example
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Example
References
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Algebra for College Students (Ch. 9, 12)
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