perform the indicated operation.
5) ( g(x) = x + 4 )
( h(x) = 2x - 4 )
find ( (g + h)(x) )
6) ( g(n) = 3n )
( h(n) = -4n - 2 )
find ( (g - h)(n) )
7) ( h(n) = 3n - 1 )
( g(n) = 2n - 2 )
find ( (h cdot g)(n) )
8) ( g(t) = t^2 - t )
find ( (g circ g)(t) )

Question

Accepted Answer

### Problem 5:
# Explanation:
## Step1: Recall the definition of function addition
The sum of two functions $(g + h)(x)$ is defined as $g(x) + h(x)$.
## Step2: Substitute the given functions
We know $g(x)=x + 4$ and $h(x)=2x-4$. So, $(g + h)(x)=g(x)+h(x)=(x + 4)+(2x-4)$.
## Step3: Simplify the expression
Combine like terms: $x+2x+4 - 4=3x$.
# Answer:
$3x$

### Problem 6:
# Explanation:
## Step1: Recall the definition of function subtraction
The difference of two functions $(g - h)(n)$ is defined as $g(n)-h(n)$.
## Step2: Substitute the given functions
We know $g(n) = 3n$ and $h(n)=-4n-2$. So, $(g - h)(n)=g(n)-h(n)=3n-(-4n - 2)$.
## Step3: Simplify the expression
Distribute the negative sign: $3n + 4n+2=7n + 2$.
# Answer:
$7n + 2$

### Problem 7:
# Explanation:
## Step1: Recall the definition of function multiplication
The product of two functions $(h\cdot g)(n)$ is defined as $h(n)\cdot g(n)$.
## Step2: Substitute the given functions
We know $h(n)=3n - 1$ and $g(n)=2n-2$. So, $(h\cdot g)(n)=h(n)\cdot g(n)=(3n - 1)(2n-2)$.
## Step3: Expand the product
Using the distributive property (FOIL method): $3n\times2n-3n\times2-1\times2n + 1\times2=6n^{2}-6n-2n + 2$.
## Step4: Combine like terms
$6n^{2}-(6n + 2n)+2=6n^{2}-8n + 2$.
# Answer:
$6n^{2}-8n + 2$

### Problem 8:
# Explanation:
## Step1: Recall the definition of function composition
The composition of a function with itself $(g\circ g)(t)$ is defined as $g(g(t))$.
## Step2: Substitute $g(t)$ into itself
First, find $g(t)=t^{2}-t$. Then, $g(g(t))=g(t^{2}-t)$.
## Step3: Substitute $x=t^{2}-t$ into $g(x)$
Since $g(x)=x^{2}-x$, we have $g(t^{2}-t)=(t^{2}-t)^{2}-(t^{2}-t)$.
## Step4: Expand the expression
Expand $(t^{2}-t)^{2}$ using the formula $(a - b)^{2}=a^{2}-2ab + b^{2}$, where $a = t^{2}$ and $b=t$. So, $(t^{2}-t)^{2}=t^{4}-2t^{3}+t^{2}$. Then, $(t^{4}-2t^{3}+t^{2})-(t^{2}-t)=t^{4}-2t^{3}+t^{2}-t^{2}+t$.
## Step5: Simplify the expression
Combine like terms: $t^{4}-2t^{3}+t$.
# Answer:
$t^{4}-2t^{3}+t$

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QUESTION IMAGE

Question

Explanation:

Problem 5:

Step1: Recall the definition of function addition

Step2: Substitute the given functions

Step3: Simplify the expression

Step1: Recall the definition of function subtraction

Step2: Substitute the given functions

Step3: Simplify the expression

Step1: Recall the definition of function multiplication

Step2: Substitute the given functions

Step3: Expand the product

Step4: Combine like terms

Answer:

Problem 6: