CHAPTER 1 FUNCTIONSAND MODELS
85. () = 1 + 3 2 − 4 .
(−) = 1+3(−) 2 −(−) 4 = 1+3 2 − 4 = ().
Since (−) = (), is an even function.
86. () = 1 + 3 3 − 5 , so
(−) = 1 + 3(−) 3 − (−) 5 = 1 + 3(− 3 ) − (− 5 )
= 1 − 3 3 + 5
Since this is neither () nor −(), the function is
neither even nor odd.
87. (i) If and are both even functions, then (−) = () and (−) = (). Now
( + )(−) = (−) + (−) = () + () = ( + )(), so + is an even function.
(ii) If and are both odd functions, then (−) = −() and (−) = −(). Now
( + )(−) = (−) + (−) = −() + [−()] = −[() + ()] = −( + )(), so + is an odd function.
(iii) If is an even function and is an odd function, then ( +)(−) = (−)+(−) = ()+[−()] = ()−(),
which is not ( + )() nor −( + )(), so + is neither even nor odd. (Exception: if is the zero function, then
+ will be odd. If is the zero function, then + will be even.)
88. (i) If and are both even functions, then (−) = () and (−) = (). Now
()(−) = (−)(−) = ()() = ()(), so is an even function.
(ii) If and are both odd functions, then (−) = −() and (−) = −(). Now
()(−) = (−)(−) = [−()][−()] = ()() = ()(), so is an even function.
(iii) If is an even function and is an odd function, then
()(−) = (−)(−) = ()[−()] = −[()()] = −()(), so is an odd function.
1.2 Mathematical Models: A Catalog of Essential Functions
1. (a) () = 3 + 3 2 is a polynomial function of degree 3. (This function is also an algebraic function.)
(b) () = cos 2 − sin is a trigonometric function.
(c) () =
√ 3
is a power function.
(d) () = 8 is an exponential function.
(e) =
√
2 + 1
is an algebraic function. It is the quotient of a root of a polynomial and a polynomial of degree 2.
(f) () = log 10 is a logarithmic function.
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SECTION 1.2 MATHEMATICAL MODELS:A CATALOG OF ESSENTIALFUNCTIONS
¤
23
2. (a) () =
3 2 + 2
is a rational function. (This function is also an algebraic function.)
(b) () = 23 is an exponential function.
(c) () =
√ + 4 is an algebraic function. It is a root of a polynomial.
(d) = 4 + 5 is a polynomial function of degree 4.
(e) () =
3
√
is a root function. Rewriting () as 13 , we recognize the function also as a power function.
(This function is, further, an algebraic function because it is a root of a polynomial.)
(f) =
1
2
is a rational function. Rewriting as −2 , we recognize the function also as a power function.
(This function is, further, an algebraic function because it is the quotient of two polynomials.)
3. We notice from the figure that and are even functions (symmetric with respect to the axis) and that is an odd function
(symmetric with respect to the origin). So (b)
= 5
must be . Since is flatter than near the origin, we must have
(c)
= 8
matched with and (a)
= 2
matched with .
4. (a) The graph of = 3 is a line (choice ).
(b) = 3 is an exponential function (choice ).
(c) = 3 is an odd polynomial function or power function (choice ).
(d) =
3
√
= 13 is a root function (choice ).
5. The denominator cannot equal 0, so 1 − sin 6= 0 ⇔ sin 6= 1 ⇔ 6=
2
+ 2. Thus, the domain of
() =
cos
1 − sin
is
| 6=
2
+ 2, an integer .