Answers:


In Fig. S1.2 , set A = 3.4, B = −π/2, C = 2π, , , F = 9/2, G = −29/10.


Let A = (2, −4), B = (−6, 3), C = (π, π ^{2} ), , F = (1/3, −19/4).




(a) Slope is −3/8 hence line is y − (−9) = (−3/8) · ( x − 4)
(b) Slope is 1 hence line is y − (−8) = 1 · ( x − (−4))
(c) y − 6 = (−8) ( x − 4)
(d) Slope is hence line is y − 3 = (−1/8)( x − 2)
(e) y = 6 x
(f) Slope is −3 hence line is y − 7 = (−3)( x −(−4))


(a) Each person has one and only one father. This is a function.
(b) Some men have more than one dog, others have none. This is not a function.
(c) Some real numbers have two square roots while others have none. This is not a function.
(d) Each positive integer has one and only one cube. This is a function.
(e) Some cars have several drivers. In a onecar family, everyone drives the same car. So this is not a function.
(f) Each toe is attached to one and only one foot. This is a function.
(g) Each rational number succeeds one and only one integer. This is a function.
(h) Each integer has one and only one successor. This is a function.
(i) Each real number has a well defined square, and adding six is a well defined operation. This is a function.



We check the first six identities.

We shall do (a), (c), (e).

(a) θ = (15/2)°
(b) θ = −60°
(c) θ = 405°
(d) θ = (405/4)°
(e) θ = (540/π)°
(f) θ = (−900/π)°

(a) θ = 13π/36 radians
(b) θ = π/18 radians
(c) θ = −5π/12 radians
(d) θ = −2π/3 radians
(e) θ = π ^{2} /180 radians
(e) θ = 157π/9000 radians

(a) f ο g ( x ) = [( x − 1) ^{2} ] ^{2} + 2[( x − 1) ^{2} ] + 3; g ο f ( x ) = ([ x ^{2} + 2 x + 3] − 1) ^{2} .
(b)
(c) f ο g ( x ) = sin([cos( x ^{2} − x )] + 3[cos( x ^{2} − x )] ^{2} ); g ο f ( x ) = cos([sin( x + 3 x ^{2} )] ^{2} − [sin( x + 3 x ^{2} ]).
(d) f ο g ( x ) = e ^{ln( x − 5)+2} ; g ο f ( x ) = ln( e ^{x +2} − 5).
(e) f ο g ( x ) = sin([ln( x ^{2} − x )] ^{2} + [ln( x ^{2} − x )]); g ο f ( x ) = ln([sin( x ^{2} + x )] ^{2} − [sin( x ^{2} + x )]).
(f) f ο g ( x ) = e ^{[ e − x 2 ] 2} ; f ο g ( x ) = e ^{−[ e x 2 ] 2} .
(g) f ο g ( x ) = [(2 x − 3)( x + 4)] · [(2 x − 3)( x + 4) + 1] · [(2 x − 3)( x + 4) + 2]; g ο f ( x ) = (2[ x ( x + 1)( x + 2)]−3)([ x ( x + 1)( x + 2)] + 4).

(a) f is invertible, with f ^{−1} ( t ) = ( t − 5) ^{1/3} .
(b) g is not invertible since g (0) = g (1) = 0.
(c) h is invertible, with h ^{−1} ( t ) = sgn t · t ^{2} .
(d) f is invertible, with f ^{−1} ( t ) = ( t − 8) ^{1/5} .
(e) g is invertible, with g ^{−1} ( t ) = −[ln t ]/3.
(f) h is not invertible, since .
(g) f is not invertible, since tan π/4 = tan 9π/4 = 1.
(h) g is invertible, with .

We will do (a), (c), (e), and (g).
(g) Not invertible.

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