Page 15 - How To Solve Word Problems In Calculus

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equation x + y = 25 defines the correspondence between x
and y, then y is not a function of x.If x = 3, for example, then
y could be 4 or −4.
Functions are usually represented symbolically by a letter
such as f or g. For convenience, function notation is often used
in calculus. In terms of the definition above, if f is a function
and x ε A, then f (x) is the unique number in B corresponding
to x.
It is not uncommon to use a letter that reminds us of
what a function represents. Thus for example, A(x) may be
used to represent the area of a square whose side is x or V(r)
may represent the volume of a sphere whose radius is r.

EXAMPLE 2
Suppose f represents the “squaring” function, i.e., the func-
tion that squares x. We write

f (x) = x 2

To compute the value of this function for a particular
value of x, simply replace x by that value wherever it appears
in the definition of the function.

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f (3) = 3 = 9
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f (−5) = (−5) = 25
f (π) = π 2
√ √
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f ( 7) = ( 7) = 7
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f (a + b) = (a + b) = a + 2ab + b 2
The domain of a function is the set of numbers for which
the function is defined. While polynomials have the set of
all real numbers as their domain, many functions must,
by their very definition, have restricted domains. For ex-
√
ample, f (x) = x has, as its domain, the set of nonnegative
real numbers. (The square root of a negative number is unde-
fined.) The domain of g(x) = 1/x is the set of all real numbers
except 0.

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