I need some clarity

[khris le]

The answer for No.12 is in red. I kind of understand the answer but not the reasoning or how to do the working for it.

This is what I've tried to do. h(x) = f(g(x))

so, (5x+1)^2 - (5x+1) = f(g(x))

Therefore g(x) = (5x+1) - (5x+1) because those were the input values into f(x)= x^2

Where did I go wrong?

 

[Dr.Peterson]

This is what I've tried to do. h(x) = f(g(x))

so, (5x+1)^2 - (5x+1) = f(g(x))

Therefore g(x) = (5x+1) - (5x+1) because those were the input values into f(x)= x^2

Where did I go wrong?

In general, not knowing what [imath]g(x)[/imath] is, but knowing that [imath]f(x) = x^2-x[/imath], we can say that [imath]f(g(x)) = (g(x))^2 - (g(x))[/imath], right?

So you are looking for one function that you can put in both places to turn it into [imath](5x+1)^2 - (5x + 1)[/imath]. Do you see that it is 5x + 1?

Otherwise, you could do it the long way: [imath]g(x) = f^{-1}(f(g(x))) =f^{-1}(h(x))[/imath], so just find the inverse of f and apply it. But that's the long way.

 

[Dr.Peterson]

Otherwise, you could do it the long way: [imath]g(x) = f^{-1}(f(g(x))) =f^{-1}(h(x))[/imath], so just find the inverse of f and apply it. But that's the long way.

I mentioned this mostly as a joke for those who are interested; but actually carrying it out, I find that there are two valid answers, and the second one can also be found in a way similar to what is expected, by simplifying h(x) first.