Hello there. To solve this question, we'll have to remember some properties about functions, compositions and products.
Given the functions:
[tex]\begin{gathered} f(x)=3x-4 \\ g(x)=x+1 \\ h(x)=2x^2-x-3 \end{gathered}[/tex]We want to find the expressions for:
(h-g)(x)
In this case, we need to calculate h(x) - g(x)
h(x) - g(x) = 2x² - x - 3 - (x + 1)
(h - g)(x) = 2x² - x - 3 - x - 1
(h - g)(x) = 2x² - 2x - 4
The domain of this function still the same, because it is a polynomial function.
(fh)(x)
We need to calculate the product between f(x) and h(x)
f(x) . h(x) = (3x - 4)(2x² - x - 3)
Apply the FOIL
(fh)(x) = 6x³ - 3x² - 9x - 8x² + 4x + 12
(fh)(x) = 6x³ - 11x² - 5x + 12
Again, the domain of this function is the same because it is a polynomial function.
(f o g)(x)
Now, we need to compose f with g. For this, we plug in the expression for g as if it was a number:
f(g(x)) = 3g(x) - 4
(f o g)(x) = 3(x + 1) - 4
(f o g)(x) = 3x + 3 - 4
(f o g)(x) = 3x - 1
The domain remains untouched.
(g o h)(x)
Same as before, compose g with h:
g(h(x)) = h(x) + 1
(g o h)(x) = 2x² - x - 3 + 1
(g o h)(x) = 2x² - x - 2
The domain is the same.
