Question 243864: FOR THE INDICATED FUNCTIONS F AND G,FIND FUNCTIONS F0G AND G0F, FIND THE DOMAINS.
F(X)=X^2;G(X)=X^3+2X+4 Answer by Theo(13342) (Show Source):
you confirm your answer is good by taking any value of x and substituting i the original equation and the final equation to see if the equations are true.
I did with x = 2 and confirmed the answers were the same so I believe the answer is good.
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to get g(f(x)):
you start with g(x) = x^3 + 2x + 4
you replace x with f(x) to get:
g(f(x)) = (f(x))^3 + 2*(f(x)) + 4
you replace f(x) with x^2 to get:
g(f(x)) = (x^2)^3 + 2*(x^2) + 4
you simplify to get:
g(f(x)) = x^6 + 2x^2 + 4
You confirm by replacing x with a random value and solving both the original equation and the final equation to see if they are the same.
I did with x = 3 and both equations came out the same so I believe the answer is good.
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The domains are all real values of x since there is no division of 0 and there are no even roots of negative numbers (square root, fourth root, etc). Those, if present, would restrict the domain.
Graph of x^2 looks like this:
Graph of x^3 + 2x + 4 looks like this:
Graph of x^6 + 4x^4 + 8x^3 + 4x^2 + 16x + 16 looks like this:
Graph of x^6 + 2x^2 + 4 looks like this:
In the last 2 graphs, the scale of the y-axis was changed from +/- 20 to +/- 100 because they become very large.
