MATH 131-505 Spring 2015
1.3
c
Wen
Liu
1.3 New Functions from Old Functions
Vertical and Horizontal Shifts: Suppose c > 0. To obtain the graph of
• y = f (x) + c, shift the graph of y = f (x) a distance c units upward.
• y = f (x) − c, shift the graph of y = f (x) a distance c units downward.
• y = f (x + c), shift the graph of y = f (x) a distance c units to the left.
• y = f (x − c), shift the graph of y = f (x) a distance c units to the right.
Vertical and Horizontal Stretching and Reflecting: Suppose c > 1. To obtain the graph of
• y = cf (x), stretch the graph of y = f (x) vertically by a factor of c
1
• y = f (x), shrink the graph of y = f (x) vertically by a factor of c
c
• y = f (cx), shrink the graph of y = f (x) horizontally by a factor of c
x
• y = f ( ), stretch the graph of y = f (x) horizontally by a factor of c
c
• y = −f (x), reflect the graph of y = f (x) about the x-axis
• y = f (−x), reflect the graph of y = f (x) about the y-axis
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MATH 131-505 Spring 2015
c
Wen
Liu
1.3
Note: The order of transformations is important. Perform the transformation in the order you would
perform the mathematical calculations for each value of x when determining the resulting y-value.
Examples: Sketch the graphs of the following functions.
√
1. y = 5 + 3 x − 2.
2. y = −(x + 1)2 − 3.
Combinations of Functions
Two functions f and g can be combined to form new functions f + g, f − g, f g, and f /g in a manner
similar to the way we add, subtract, multiply, and divide real numbers.
Let the domain of f to be A and the domain of g to be B.
(a) (f ± g)(x) = f (x) ± g(x)
The domain of f + g is the intersection A ∩ B because both f (x) and g(x) have to be defined.
Example 3: (p. 41) Find domains of (f ± g) (x) for f (x) =
√
√
x and g(x) = 2 − x.
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MATH 131-505 Spring 2015
1.3
c
Wen
Liu
f
f (x)
(b) (f g)(x) = f (x)g(x),
(x) =
g
g(x)
The domain of f g is A ∩ B.
The domain of f /g is {x ∈ A ∩ B | g(x) 6= 0}.
Example 4: (p. 41) Find domains of f g and f /g for f (x) = x2 and g(x) = x − 1.
Given two functions f and g, the composite function f ◦ g (also called the composition of f and
g) is defined by
(f ◦ g)(x) = f (g(x))
The domain of f ◦ g is the set of all x in the domain of g such that g(x) is in the domain of f . In
other words, (f ◦ g)(x) is defined whenever both g(x) and f (g(x)) are defined.
√
√
Examples 5: (p. 42) Given f (x) = x, g(x) = 2 − x, and h(x) = x2 . Find each function and its
domain.
(a) f ◦ g. Use it to compute f (g(1)).
(b) g ◦ g. Use it to compute (g ◦ g)(0)
(c) f ◦ g ◦ h
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MATH 131-505 Spring 2015
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c
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Liu
Decomposing a function: decompose a complicated function into simpler ones, as in the following
example.
Examples:
√
6. Express the function u(t) = x2 + 1 in the form of f ◦ g.
7. If g(x) = 2x + 1 and h(x) = 4x2 + 4x + 7, find a function f such that f ◦ g = h.
8. If f (x) = 6x + 8 and h(x) = 6x2 + 6x + 2, find a function g such that f ◦ g = h.
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