# Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.?

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1.) f(x)=x-7/x+3 and g(x) = -3x-7/x-1

2.)Verify the identity. cos 4x + cos 2x = 2 – 2 sin^2 2x – 2 sin^2 x

• 1)

You need to put parenthesis in your question. Did you mean g(x) = (-3x-7)/(x-1)?

f(x) = (x-7) /(x+3)

f(g(x)) = f [ (-3x-7) /(x-1)]

Replace x with (-3x-7)/(x-1) in f

Evaluate (-3x-7)/(x-1) by replacing x with (x-7)/(x+3)

evaluate the numerator (-3x-7):

– 3 (x-7)/(x+3) -7 = (-3x+21)/(x+3) – 7

= [(-3x+21) -(7x+21)] /(x+3) = (-3x+21-7x-21)/(x+3) = -10x/(x+3) ——(1)

evaluate the denominator (x+3)

(-3x-7) /(x+3) +3

= [ (-3x-7) + 3(x-1)] /(x+3) = (-3x-7+3x-3)/(x+3) =-10/(x+3) ——–(2)

(1)/(2) = -10x/-10 = x

We have shown that f(g(x)) = x

————————————————–

g(x) = [(-3x-7) /(x-1)]

numerator of g(f(x)) = [ -3( (x-7)/(x+3) – 7] = [-3x+21-7(x+3)]/(x-1) = (-3x+21-7x-21)/(x-1) = -10x/(x-1)

denominator of g(f(x)) = (x-7)/(x+3)-1 = ((x-7)-x-3)/(x-1) = -10/(x-1)

divide = -10x/(x-1)/-10/(x-1) = x

f(g(x)) = g(f(x))

——————————————————————————-

2)

cos 4x +cos 2x = 2 cos ^2 2x -1 + 2 cos^2 x -1

= 2(1 – sin^2 2x ) -1 +cos 2x

=1-2sin^2 2x + 1-2sin^2 x

=2 – 2sin^2 2x – 2 sin^2 x

=

• Need ()

assuming I use the right ()

f(x)= (x-7)/(x+3)

g(x) = (-3x-7)/(x-1)

f(g(x)) = ((-3x-7)/(x-1)-7)/((-3x-7)/(x-1)+3 )

distribute then simplify

= x

g(f(x)) = (-3(x-7)/(x+3)-7)/((x-7)/(x+3)-1)

=x

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