Question

Q.1

Answer 1

The question regards composite functions. A composite function is a function composed of more than one function. Sorry for saying the word function so many times there, it's just what it is...

The phrase f(g(x)) means 'perform g on an input x, then perform f on the result'. You can then see that there are many options for f(x) and g(x) here, in fact an infinite number of one were to be ridiculous about it.

However a sensible choice might be g(x) = x^2, and f(x) = 2/x + 9. Checking:

g(x) = x^2
f(g(x)) = 2/(x^2) + 9

That is the first question dealt with. Next up is Q2. It is relatively simple to show that these functions are inverses. If you start with a value x, apply a function and then apply the function's inverse, you should return to the same starting value x. To take a common example, within a certain domain, sin^-1(sin(x)) = x.

f(g(x)) = (sqrt(3+x))^2 - 3 = 3 + x - 3 = x

g(f(x)) = sqrt(x^2 - 3 + 3) = sqrt(x^2) = x

A final note is that this is only true for a certain domain, that is x <= 0. This is because y = x^2 is a many-to-one function, so unrestricted it does not have an inverse. Take the example to illustrate this:

If x = -2, f(x) = (-2)^2 - 3 = 4 - 3 = 1

Then g(f(x)) =sqrt(1 + 3) = sqrt(4) = 2 (principal value).

However the question isn't testing knowledge of that.

I hope this helps you :)