In applications, the symbols used for the independent and dependent variables are often based on common usage. So, rather than using y = f ( x ) to represent a function, an applied problem might use C = C ( q ) to represent the cost C of manufacturing q units of a good. Because of this, the inverse notation f − 1 used in a pure mathematics problem is not used when finding inverses of applied problems. Rather, the inverse of a function such as C = C ( q ) will be q = q ( C ) . So C = C ( q ) is a function that represents the cost C as a function of the number q of units manufactured, and q = q ( C ) is a function that represents the number q as a function of the cost C . Problems 91-94 illustrate this idea. Height and Head Circumference The head circumference C of a child is related to the height H of the child (both in inches) through the function H ( C ) = 2.15 C − 10.53 (a) Express the head circumference C as a function of height H . (b) Verify that C = C ( H ) is the inverse of H = H ( C ) by showing that H ( C ( H ) ) = H and C ( H ( C ) ) = C . (c) Predict the head circumference of a child who is 26 inches tall.
In applications, the symbols used for the independent and dependent variables are often based on common usage. So, rather than using y = f ( x ) to represent a function, an applied problem might use C = C ( q ) to represent the cost C of manufacturing q units of a good. Because of this, the inverse notation f − 1 used in a pure mathematics problem is not used when finding inverses of applied problems. Rather, the inverse of a function such as C = C ( q ) will be q = q ( C ) . So C = C ( q ) is a function that represents the cost C as a function of the number q of units manufactured, and q = q ( C ) is a function that represents the number q as a function of the cost C . Problems 91-94 illustrate this idea. Height and Head Circumference The head circumference C of a child is related to the height H of the child (both in inches) through the function H ( C ) = 2.15 C − 10.53 (a) Express the head circumference C as a function of height H . (b) Verify that C = C ( H ) is the inverse of H = H ( C ) by showing that H ( C ( H ) ) = H and C ( H ( C ) ) = C . (c) Predict the head circumference of a child who is 26 inches tall.
Solution Summary: The author explains that the head circumference C of a child is related to the height H of the child through the function.
In applications, the symbols used for the independent and dependent variables are often based on common usage. So, rather than using
to represent a function, an applied problem might use
to represent the cost
of manufacturing q units of a good. Because of this, the inverse notation
used in a pure mathematics problem is not used when finding inverses of applied problems. Rather, the inverse of a function such as
will be
. So
is a function that represents the cost
as a function of the number
of units manufactured, and
is a function that represents the number
as a function of the cost
. Problems 91-94 illustrate this idea.
Height and Head Circumference The head circumference
of a child is related to the height
of the child (both in inches) through the function
(a) Express the head circumference
as a function of height
.
(b) Verify that
is the inverse of
by showing that
and
.
(c) Predict the head circumference of a child who is 26 inches tall.
Heller Manufacturing has two production facilities that manufacture baseball gloves. Production costs at the two facilities differ because of varying labor rates, local property taxes, type of equipment,
capacity, and so on. The Dayton plant has weekly costs that can be expressed as a f
a function of the number of gloves produced
TCD(X)=x²-x*4
where x is the weekly production volume in thousands of units and TCDX)Is the cost in thousands of dollars. The Hamiton plant's weekly production costs are given by
TCH(Y) = y² + 2Y+8
where Y is the weekly production volume in thousands of units and TCH(Y) is the cost in thousands of dollars. Heller Manufacturing would like to produce 9,000 gloves per week at the lowest possible
cost.
(a) Formulate a mathematical model that can be used to determine the optimal number of gloves to produce each week at each facility.
min
st.
X, Y 20
(b) Use Excel Solver or LINGO to find the solution to your mathematical model to determine the optimal number of…
I am not sure I am understanding the one-to-one part. If the functions are equal to one another, wouldn't that mean that they aren't one-to-one
20) What is the inverse of the function 4y= 2x-8 ?
Chapter 5 Solutions
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