Expert: Sherman D. - 10/4/2009

Question
my problems (set 2):
2. Let g(x)= radical(x). Find:
(a) g(5s+2)
(b) g(radical(x)+2)
(c) 3g(5x)
(d) 1/g(x)
(e) g(g(x))
(f) g²(x)
(g) g(1/radical(x))
(h) g((x-1)²)
3. Given that f(-1)=4, f(2)=5, g(-1)=3, and g(2)=-1, find:
(a) (f-g)(-1)
(b) (f*g)(-1)
(c) (f/g)(2)
(d) (f of g)(2)
4. Let f(x)=3/x. Find:
(a) f(1/x)+(1/f of x)
(b) f(x²)-f²(x)
16. In Exercise 16, find f(x+h)-f(x)/h and simplify as much as possible. The question is f(x)=1/x².
9. If f(x)=2x²+4 and g(x)=x-3, which number satisfies f(x)=(f of g)(x)?
[A] ¾
[B] 3/2
[C] 5
[D] 4
32. For question 32, find the standard equation of the circle satisfying the given conditions. A diameter has endpoints (6,1) and (-2,3).
25. In Exercise 25, find f(x+h)-f(x)/h and simplify as much as possible. The question is f(x)=x².

Answer
  my problems (set 2):
2. Let g(x)= radical(x). Find:
(a) g(5s+2) = sqrt(5s + 2)

(b) g(radical(x)+2) = sqrt(sqrt(x) + 2))

(c) 3g(5x) = 3sqrt(5x)

(d) 1/g(x) = 1/sqrt(x) = sqrt(x)/x

(e) g(g(x)) = sqrt(sqrt(x))

(f) g²(x) = x

(g) g(1/radical(x)) = sqrt(1/sqrt(x)) = 1/(4thrt(x))

(h) g((x-1)²) = sqrt((x - 1)^2) = x - 1

--------------------------------------------------------------

3. Given that f(-1)=4, f(2)=5, g(-1)=3, and g(2)=-1, find:

(a) (f-g)(-1) = 4 - 3 = 1

(b) (f*g)(-1) = 4 * 3 = 12

(c) (f/g)(2) = 5/(-1) = -5

(d) (f of g)(2) =

assuming this is a straight line

(-1,4) and (2,5)
m = (5 - 4)/(2 - (-1))
m = 1/3

(-1,4), m = (1/3)
4 = (-1/3) + b
b = (12/3) + (1/3)
b = (13/3)

f(x) = (1/3)x + (13/3)

(-1,3) and (2,-1)
m = (-1 - 3)/(2 - (-1))
m = (-4)/(3)

(-1,3), m = (-4/3)
3 = (4/3) + b
b = (9/3) - (4/3)
b = (5/3)

g(x) = (-4/3)x + (5/3)

so to have (f o g)(2) or f(g(x))(2)

g(2) = (-4/3)(2) + (5/3)
g(2) = (-8/3) + (5/3)
g(2) = (-3/3)
g(2) = -1

f(-1) = 4

by the looks of it, i probably didn't even have to work it out.

---------------------------------------------------------------

4. Let f(x)=3/x. Find:
(a) f(1/x)+(1/f of x)
f(1/x) = 3/(1/x) = 3x

if by 1/f of x, you mean 1/f(x) = 1/(3/x) = (x/3)

f(1/x) + (1/f(x)) = 3x + (x/3) = (9x + x)/3 = (10/3)x

(b) f(x²) - f²(x)

f(x^2) = 3/(x^2)
f(x)^2 = (3/x)^2 = 9/(x^2)

f(x^2) - f(x)^2 = 3/(x^2) - 9/(x^2) = -6/(x^2)

16. In Exercise 16, find f(x+h)-f(x)/h and simplify as much as possible. The question is f(x)=1/x².

f(x) = 1/(x^2)

f(x + h) = 1/((x + h)^2)

((1/(x^2 + 2xh + h^2)) - (1/(x^2)))/h
((x^2 - x^2 - 2xh - h^2)/((x^2)(x^2 - 2xh + h^2)))/h
(-2xh - h^2)/((hx^2)(x^2 - 2xh + h^2))
(h(-2x - h))/((hx^2)(x^2 - 2xh + h^2))
(-2x - h)/(x^4 - 2x^3h + x^2h^2)

--------------------------------------------------------------

9. If f(x)=2x²+4 and g(x)=x-3, which number satisfies f(x)=(f of g)(x)?
[A] ¾
[B] 3/2
[C] 5
[D] 4

(f of g)(x) or f(g(x))(x)

f(x - 3) = 2(x - 3)^2 + 4
f(g(x))(x) = 2(x^2 - 6x + 9) + 4
f(g(x))(x) = 2x^2 - 12x + 18 + 4
f(g(x))(x) = 2x^2 - 12x + 22

2x^2 + 4 = 2x^2 - 12x + 22
12x = 18
x = 18/12
x = 3/2

ANS : B.

--------------------------------------------------

32. For question 32, find the standard equation of the circle satisfying the given conditions. A diameter has endpoints (6,1) and (-2,3).

D = sqrt((-2 - 6)^2 + (3 - 1)^2)
D = sqrt((-8)^2 + (2)^2)
D = sqrt(64 + 4)
D = sqrt(68)

D = 2sqrt(17)

r = sqrt(17)

now we just find the center

M = ((-2 + 6)/2),((3 + 1)/2)
M = (4/2),(4/2)
M = 2,2

ANS : (x - 2)^2 + (y - 2)^2 = 17

---------------------------------------------------------

25. In Exercise 25, find f(x+h)-f(x)/h and simplify as much as possible. The question is f(x)=x².

(x^2 + 2xh + h^2 - x^2)/h = (2xh + h^2)/h = 2x + h