Question 1201999
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Normally I would point out that the rule of this website is "one question per post". 


However, these questions are fairly short so it's not the end of the world to have them all together like this.


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Problem 1


Method 1
f(x) = 2x
f(2) = 2*2
f(2) = 4
and
h(x) = x^2-1
h(2) = 2^2-1
h(2) = 4-1
h(2) = 3
those results lead to
(fh)(2) = f(2)*h(2)
(fh)(2) = 4*3
(fh)(2) = 12


Method 2
(fh)(x) = f(x)*h(x)
(fh)(x) = ( f(x) )*( h(x) )
(fh)(x) = ( 2x )*( x^2-1 )
(fh)(x) = 2x^3-2x
(fh)(2) = 2*2^3-2*2
(fh)(2) = 2*8-4
(fh)(2) = 16-4
(fh)(2) = 12



Answer: <font color=red size=4>12</font>


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Problem 2


Method 1
g(x) = x-5
g(-2) = -2-5
g(-2) = -7
and
h(x) = x^2-1
h(-2) = (-2)^2-1
h(-2) = 4-1
h(-2) = 3
so,
(gh)(-2) = g(-2)*h(-2)
(gh)(-2) = -7*3
(gh)(-2) = -21


Method 2
(gh)(x) = g(x)*h(x)
(gh)(x) = ( g(x) ) * ( h(x) )
(gh)(x) = ( x-5 ) * ( x^2-1 )
(gh)(x) = x( x^2-1 ) - 5( x^2-1 )
(gh)(x) = x^3-x -5x^2+5 
(gh)(x) = x^3-5x^2-x+5 
(gh)(-2) = (-2)^3-5(-2)^2-(-2)+5 
(gh)(-2) = -8-5*(4)+2+5 
(gh)(-2) = -8-20+2+5
(gh)(-2) = -21



Answer: <font color=red size=4>-21</font>


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Problem 3


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



Answer: <font color=red size=4>(2x)/(x^2-1)</font>


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Problem 4


Method 1
h(x) = x^2-1
h(0) = 0^2-1
h(0) = 0-1
h(0) = -1
and
g(x) = x-5
g(0) = 0-5
g(0) = -5
therefore,
(h/g)(x) = ( h(x) )/( g(x) )
(h/g)(0) = ( h(0) )/( g(0) )
(h/g)(0) = ( -1 )/( -5 )
(h/g)(0) = 1/5



Method 2
(h/g)(x) = ( h(x) )/( g(x) )
(h/g)(x) = ( x^2-1 )/( x-5 )
(h/g)(0) = ( 0^2-1 )/( 0-5 )
(h/g)(0) = ( -1 )/( -5 )
(h/g)(0) = 1/5



Answer: <font color=red size=4>1/5</font>


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Problem 5


g(x) = x-5
f(x) = 2x
(g/f)(x) = ( g(x) )/( f(x) )
(g/f)(x) = ( x-5 )/( 2x )


Answer: <font color=red size=4>( x-5 )/( 2x )</font>

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Summary:<ol><li><font color=red size=4>12</font></li><li><font color=red size=4>-21</font></li><li><font color=red size=4>(2x)/(x^2-1)</font></li><li><font color=red size=4>1/5</font></li><li><font color=red size=4>( x-5 )/( 2x )</font></li></ol>
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