Understanding Composition Function (fg)(-1) for f(x)=x-3 & g(x)=x^2-8x+15?
Can someone help explain how to do the following composition function to me? (or at least get me started)
Find the value of (fg)(-1) if f(x) = x - 3 and g(x) = x^2 - 8x + 15
Is it simply plugging in the -1 into the f(x) then using the result as the input value for g(x)? Ex:
$$f(-1) = (-1) - 3$$ $$f(-1) = -4$$ $$g(-4) = (-4)^2 - 8(-4) + 15$$ $$g(-4) = 16 + 32 + 15$$ $$g(-4) = 63$$
or would it be more like this:
$$(fg)(-1) = (x-3)(x^2 - 8x + 15)$$ $$(fg)(-1) = ((-1) - 3)((-1)^2 - 8(-1) + 15)$$ $$(fg)(-1) = ((-1) - 3)(1 + 8 + 15)$$ $$(fg)(-1) = ((-1) - 3)(24)$$ $$(fg)(-1) = -24 - 72 = -96?$$
$\endgroup$ 11 Answer
$\begingroup$Judging from the additional context provided in a comment$^1$, the notation means pointwise multiplication, that is $$(fg)(x) := f(x) \cdot g(x)\\ (fg)(-1) = f(-1)\cdot g(-1) = -4\cdot 24 = -96$$ Similarly, you can define other pointwise operations as $$\begin{align*} (f/g)(x) &:= \frac{f(x)}{g(x)}\\ (f+g)(x) &:= f(x) + g(x)\\ (f-g)(x) &:= f(x) - g(x) \end{align*}$$ And composition is denoted by $$(f\circ g)(x) := f(g(x))$$
$^1$The answer was subsequently deleted, here is a copy of the comment for the <10k users:
@AlexR I wasn't sure if the notation (fg)(-1) was the (f o g) notation or not. The problem next to it is (g/f)(2)... So then by your hint it would be the following?: g(-1) = (-1)^2 - 8(-1) + 15 = 25; f(25) = (25-3) = 22
