The following functions: f(x)=the square root of [(4-7x)\(4-x)] and g(x)= 2x^2+3. What is f composed with g?

Question

And what is g composed with f?

I tried to complete these but ran into conflicts while simplying.

Answer 1

To get f composed with g; plug in g(x) for every instance of x in f(x).

f(g(x)) = sqrt [ (4 - 7g(x)) / (4 - g(x)) ]
f(g(x)) = sqrt [ (4 - 7{2x^2 + 3}) / (4 - {2x^2 + 3}) ]
f(g(x)) = sqrt [ (4 - 14x^2 - 21) / (4 - 2x^2 - 3) ]
f(g(x)) = sqrt [ (-17 - 14x^2) / (1 - 2x^2) ]

Just because you might get a different answer in your book, let's rationalize the denominator.

f(g(x)) = sqrt (-17 - 14x^2) / sqrt (1 - 2x^2)

Multiply top and bottom by sqrt (1 - 2x^2) to get

f(g(x)) = sqrt [ (-17 - 14x^2) (1 - 2x^2) ] / (1 - 2x^2)
f(g(x)) = sqrt [ (-17 + 34x^2 - 14x^2 + 28x^4) ] / (1 - 2x^2)
f(g(x)) = sqrt [ (-17 + 20x^2 + 28x^4) ] / (1 - 2x^2)

Since you were interested in g composed with f, I'll solve that too.

g(f(x)) = 2[f(x)]^2 + 3
Notice that f(x) is sqrt[junk], so [f(x)]^2 would just be [junk].

g(f(x)) = 2[ (4 - 7x) / (4 - x) ] + 3
g(f(x)) = [2(4 - 7x)] / (4 - x) + 3

Put them under a common denominator, to get

g(f(x)) = [2(4 - 7x)] / (4 - x) + 3(4 - x)/(4 - x)
g(f(x)) = {[2(4 - 7x)] + [3(4 - x)]} / (4 - x)
g(f(x)) = {8 - 14x + 12 - 3x} / (4 - x)
g(f(x)) = {20 - 17x} / (4 - x)

Answer 2

f(x)=the square root of [(4-7x)\(4-x)]
g(x)= 2x^2+3
fg(x),
so substitute g(x) into f(x),

square root of [4-7(2x^2+3)\4-(2x^2+3)]
=square root of [(4-14x^2-21)\(4-2x^2-3)]
=square root of [(-14x^2-17)\(1-2x^2)]
thats the furthest i can simplify..i don't think can simplify it anymore

for gf(x),
substitute f(x) into g(x),
giving u,
=2[(4-7x)\(4-x)]+3 {square root canceled due to ^2}
=(8-14x)\(4-x)+3
=(8-14x)\(4-x)+3(4-x)\(4-x) {change to same denominator}
=(8-14x+12-3x)\(4-x)
=20-17x\4-x

hopefully my answers are correct.You should include the answers while asking maths questions next time ;)

Answer 3

g(f(x)) = g{[(4-7x)/(4-x)]^(1/2)} = 2[(4-7x)/(4-x)] +3

= (8 - 14x)/(4-x) + (12-3x)/(4-x) = (20 - 17x)/(4-x)