Question

homework assignment 3.4 composition of functions
due sep 19 by 11:59pm points 11 submitting an external tool
homework assignment 3.4 composition of functions
score: 0/11 answered: 0/11
question 3
use the given functions to find, simplify, and identify the domain of the following combinations.
f(x)=\frac{-7x + 5}{x - 1} and g(x)=\frac{-3x + 4}{x + 9}
(f + g)(x)=
domain of (f + g)(x):
(f - g)(x)=
domain of (f - g)(x):
(fg)(x)=
domain of (fg)(x):
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Explanation:

Step1: Find (f + g)(x)

Add the two functions:
\[

$$\begin{align*} (f + g)(x)&=\frac{-7x + 5}{x - 1}+\frac{-3x + 4}{x + 9}\\ &=\frac{(-7x + 5)(x + 9)+(-3x + 4)(x - 1)}{(x - 1)(x + 9)}\\ &=\frac{-7x^{2}-63x+5x + 45-3x^{2}+3x + 4x-4}{(x - 1)(x + 9)}\\ &=\frac{-10x^{2}-51x + 41}{(x - 1)(x + 9)} \end{align*}$$

\]

Step2: Find the domain of (f + g)(x)

The domain is all real - numbers except the values that make the denominator zero. Set \((x - 1)(x + 9)=0\), we get \(x = 1\) and \(x=-9\). So the domain is \(\{x\in\mathbb{R}|x
eq1,x
eq - 9\}\).

Step3: Find (f - g)(x)

Subtract the two functions:
\[

$$\begin{align*} (f - g)(x)&=\frac{-7x + 5}{x - 1}-\frac{-3x + 4}{x + 9}\\ &=\frac{(-7x + 5)(x + 9)-(-3x + 4)(x - 1)}{(x - 1)(x + 9)}\\ &=\frac{-7x^{2}-63x+5x + 45 + 3x^{2}-3x-4x + 4}{(x - 1)(x + 9)}\\ &=\frac{-4x^{2}-65x + 49}{(x - 1)(x + 9)} \end{align*}$$

\]

Step4: Find the domain of (f - g)(x)

Set the denominator \((x - 1)(x + 9)=0\), we get \(x = 1\) and \(x=-9\). So the domain is \(\{x\in\mathbb{R}|x
eq1,x
eq - 9\}\).

Step5: Find (fg)(x)

Multiply the two functions:
\[

$$\begin{align*} (fg)(x)&=\frac{-7x + 5}{x - 1}\cdot\frac{-3x + 4}{x + 9}\\ &=\frac{(-7x + 5)(-3x + 4)}{(x - 1)(x + 9)}\\ &=\frac{21x^{2}-28x-15x + 20}{(x - 1)(x + 9)}\\ &=\frac{21x^{2}-43x + 20}{(x - 1)(x + 9)} \end{align*}$$

\]

Step6: Find the domain of (fg)(x)

Set the denominator \((x - 1)(x + 9)=0\), we get \(x = 1\) and \(x=-9\). So the domain is \(\{x\in\mathbb{R}|x
eq1,x
eq - 9\}\).

Answer:

\((f + g)(x)=\frac{-10x^{2}-51x + 41}{(x - 1)(x + 9)}\)
Domain of \((f + g)(x)\): \(\{x\in\mathbb{R}|x
eq1,x
eq - 9\}\)
\((f - g)(x)=\frac{-4x^{2}-65x + 49}{(x - 1)(x + 9)}\)
Domain of \((f - g)(x)\): \(\{x\in\mathbb{R}|x
eq1,x
eq - 9\}\)
\((fg)(x)=\frac{21x^{2}-43x + 20}{(x - 1)(x + 9)}\)
Domain of \((fg)(x)\): \(\{x\in\mathbb{R}|x
eq1,x
eq - 9\}\)