Question

X

f(x) = x+1¹ g(x) =
1'
(a)
(f + g)(x) =
(b) (f- g)(x) =
(c) (fg)(x) =
1
+7.
(d) (f/g)(x) =
X
What is the domain of f/g? (Enter your answer using interval notati

Answer 1

`\textsf{a)}\quad (f+g)(x)=(x^5+x+1)/(x^5+x^4)`

`\textsf{b)}\quad (f-g)(x)=(x^5-x-1)/(x^5+x^4)`

`\textsf{c)}\quad (fg)(x)=(1)/(x^4+x^3)`

`\textsf{d)}\quad (f/g)(x)=(x^5)/(x+1)`

The domain of (f/g)(x) is (-∞, -1) ∪ (-1, 0) ∪ (0, ∞).

Explanation:

Given functions:

`f(x)=(x)/(x+1)`

`g(x)=(1)/(x^4)`

a) To calculate (f + g)(x), we need to add the functions f(x) and g(x) together.

`\begin{aligned}(f+g)(x)&=f(x)+g(x)\\\\&=(x)/(x+1)+(1)/(x^4)\\\\&=(x\cdot x^4)/((x+1)\cdot x^4)+(1\cdot (x+1))/(x^4\cdot (x+1))\\\\&=(x^5)/(x^4(x+1))+((x+1))/(x^4(x+1))\\\\&=(x^5+(x+1))/(x^4(x+1))\\\\&=(x^5+x+1)/(x^5+x^4)\end{aligned}`

b) To calculate (f - g)(x), we need to subtract function g(x) from function f(x):

`\begin{aligned}(f-g)(x)&=f(x)-g(x)\\\\&=(x)/(x+1)-(1)/(x^4)\\\\&=(x\cdot x^4)/((x+1)\cdot x^4)-(1\cdot (x+1))/(x^4\cdot (x+1))\\\\&=(x^5)/(x^4(x+1))-((x+1))/(x^4(x+1))\\\\&=(x^5-(x+1))/(x^4(x+1))\\\\&=(x^5-x-1)/(x^5+x^4)\end{aligned}`

c) To calculate (fg)(x), we need to multiply function f(x) by function g(x):

`\begin{aligned}(fg)(x)&=f(x)\cdot g(x)\\\\&=(x)/(x+1)\cdot (1)/(x^4)\\\\&=(x)/((x+1)x^4)\\\\&=(x)/(x^5+x^4)\\\\&=(1)/(x^4+x^3)\end{aligned}`

d) To calculate (f/g)(x), we need to divide function f(x) by function g(x):

`\begin{aligned}(f/g)(x)&=(f(x))/(g(x))\\\\&=((x)/(x+1))/((1)/(x^4))\\\\&=(x)/(x+1)\cdot (x^4)/(1)\\\\&=(x^5)/(x+1)\end{aligned}`

A rational function is undefined when its denominator is equal to zero.

Therefore:

• The domain of f(x) is (-∞, -1) ∪ (-1, ∞).
• The domain of g(x) is (-∞, 0) ∪ (0, ∞).

To determine the domain of (f/g)(x), combine the domains of f(x) and g(x).

Therefore, the domain of (f/g)(x) is:

• (-∞, -1) ∪ (-1, 0) ∪ (0, ∞).