COT 3100 Homework #3 Solutions Fall ’00
1)
Find the inverse of each of the following functions: (The domain is the set of real
number unless otherwise specified.)
a) f(x) = 12x –7
x = 12f
1
(x) – 7
12f
1
(x) = (x + 7)
f
1
(x) = (x + 7)/12, domain and range here are all reals.
b) f(x) = (3x+1)/(x2), for x
≠
2
x = (3f
1
(x) + 1)/( f
1
(x) – 2)
x( f
1
(x) – 2) = 3f
1
(x) + 1
x f
1
(x) – 2x = 3f
1
(x) + 1
x f
1
(x) – 3f
1
(x) = (1 + 2x)
f
1
(x)[x – 3] = (1 + 2x)
f
1
(x) = (2x + 1)/(x – 3), domain is all reals except x=3 and the range is all reals
except y=2. (Drawing out the graph shows that y=2 is an
asymptote.)
c) f(x) = 2
x+4
x = 2
f1(x) + 4
log
2
x = f
1
(x) + 4
f
1
(x) = log
2
x – 4, domain is all reals x > 0, and the range is all reals.
2)
Compute each of the following function compositions:
a) f(x) = 3x+7, g(x)=x
2
, compute g(f(x)).
g(f(x)) = g(3x+7)
= (3x+7)
2
= 9x
2
+ 42x + 49
b) f(x) = 2
3x – 2
, g(x)=(x+2)/3, compute f(g(x)).
f(g(x) = f((x+2)/3)
= 2
3(x+2)/3 – 2
= 2
x + 2 – 2
= 2
x
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c) f(x) = (4x
2
– 5)/(3x+2), g(x) = x – 4, compute f(g(x)).
f(g(x)) = f(x – 4)
= (4(x – 4)
2
– 5)/(3(x – 4) +2)
= (4x
2
– 32x + 64 – 5)/(3x – 12 + 2)
= (4x
2
– 32x + 59)/(3x – 10)
3)
Define f
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 Fall '09
 f1, smallest value

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