Question

If f(x) = 3x - 4, g(x) = x + 1, and h(x) = 2 - x - 3, write the function rule representing the result of each function operation. State any restrictions on the domain.

Answer 1

Hello there. To solve this question, we'll have to remember some properties about functions, compositions and products.

Given the functions:

`\begin{gathered} f(x)=3x-4 \\ g(x)=x+1 \\ h(x)=2x^2-x-3 \end{gathered}`

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.