I'm trying to make sure I got this question right. I'm supposed to find (g * f)(x) when f(x)x^2-2x and g(x)=√(x-15).
This is what I've got:
(g * f)(x) = g(f(x))
√ (x^2-2x)-15=
√(x^2-2x-15)=
√(x-5)(x+3)
Hmmm.
(g*f)(x) = g(x)*f(x)
= √(x-15)(x^2-2x)
The composite function
(g◦f)(x) = g(f(x))
= √(f-15)
= √(x^2-2x-15)
Your factoring is correct, but I see no advantage to it. It ought also to be written as √((x-5)(x+3)) to avoid confusion with
√(x-5) * (x+3)
since usually parentheses delimit arguments of functions.
Alright, so if the question is asking for a * then I multiply both functions together, but if its asking for a ◦ then I plug in one function the the other?
ps,
Thanks for your help!
To find (g * f)(x), you need to evaluate g(f(x)). Let's substitute f(x) into g(x):
g(f(x)) = √(f(x) - 15)
Now substitute f(x) with its given expression:
g(f(x)) = √((x^2 - 2x) - 15)
Simplify the expression inside the square root:
g(f(x)) = √(x^2 - 2x - 15)
At this point, it seems like you made an error in your calculation. The expression is not equal to √((x - 5)(x + 3)). Instead, it represents a quadratic equation. Let's factor it correctly:
g(f(x)) = √((x - 5)(x + 3))
The correct expression for (g * f)(x) is:
(g * f)(x) = √((x - 5)(x + 3))