Let f(x) = x^2 − 3x + 2. a. give a restricted domain for f where it is invertible. b. find the inverse

-3

Step-by-step explanation:

G(-1)=3*-1=-3

Was there more?

1) f(g(x))=x

g(f(x))=x

2) f(x) and g(x) are inverses of each other.

Step-by-step explanation:

The given functions are :

[tex]f(x) = 3x - 2[/tex]

and

[tex]g(x) = \frac{x + 2}{3}[/tex]

1) We want to find:

[tex]f(g(x)) \: and \: g(f(x))[/tex]

This implies that:

[tex]f(g(x)) = f( \frac{x + 2}{3})[/tex]

We substitute into f(x) to get:

[tex]f(g(x)) = 3( \frac{x + 2}{3}) - 2 \\f(g(x)) = x + 2- 2 \\ f(g(x)) = x[/tex]

Also,

[tex]g(f(x)) = g(3x - 2) \\ g(f(x)) = \frac{3x - 2 + 2}{3} \\ g(f(x)) = \frac{3x}{3} \\ g(f(x)) = x[/tex]

2) Since the composition of the two functions:

[tex]f(g(x)) = g(f(x)) = x[/tex]

f(x) and g(x) are inverses of each other.

10

Step-by-step explanation:

H(3)=3^2+1=10

A

f(g(x)) = f([tex]\frac{x+2}{3}[/tex]) = 3([tex]\frac{x+2}{3}[/tex]) - 2 = x + 2 - 2 = x

g(f(x)) = g(3x - 2) = [tex]\frac{3x-2+2}{3}[/tex] = [tex]\frac{3x}{3}[/tex] = x

B

Since both composite functions f(g(x)) and g(f(x)) equal x

This indicates that the functions f(x) and g(x) are inverse functions

       

First, I should point out that g(x) should be written as g(x)=(x+2)/3, otherwise the problem is confusing.  

[tex]f(x)=3x-2 \enspace g(x)=\frac{x+2}{3}[/tex]

(A) [tex]f(g(x))=3(\frac{x+2}{3})-2=x\\g(f(x))=\frac{3x-2+2}{3}=x[/tex]

(B) Since [tex]f(g(x))=x[/tex] and [tex]g(f(x))=x[/tex], it holds that

[tex]f(g(x))=g(f(x))[/tex] for all x. This means the composed functions are *identical*

The composition of the functions is g(f(x))

Step-by-step explanation:

* Lets revise what is the meaning of composite functions

- A composite function is created when one function is substituted into

  another function

- Ex: f(g(x)) is the composite function formed when g(x) is substituted for

  x in f(x)

* Lets solve the problem

- Sally earns a 2% commission on total sale over $5000

- Which is paid as a bonus at the end of the year

- Let her total sales be represented by x

- f(x) = x - 5000 and g(x) = 0.02 x

- We need to find the suitable composite function which represents her

   bonus at the end of the year

∵ She earns a 2% commission on total sale over $5000

∵ Her total sale is $x

- At first subtract from x the $5000, then multiply the answer by 2%

- That mean she earns 2% of (x - 5000)

∵ 2% = [tex]\frac{2}{100}[/tex] = 0.02

∴ She earns 0.02(x - 5000)

- Lets find the composite functions which give us 0.02(x - 5000)

∵ f(x) = x - 5000

∵ g(x) = 0.02 x

- Substitute x in g(x) by f(x)

∴ g(f(x)) = 0.02 (x - 5000)

* The composition of the functions is g(f(x))

The difference between the microfilaments, intermediate filaments, and microtubules are stated below.

Explanation:

Microfilaments:They are made up of two chains made up of monomeric globular proteins called actin. The chains are coiled around each other.They have a diameter of about 7 nanometre.They help in cellular movement.Intermediate Filaments:They form strands which are made up of fibrous proteins like keratin, vimentin, desmin.They have a diameter which ranges from 8 nanometre to 10 nanometre.They have structural function and are required to maintain the cell shape and organelle position.Microtubules:They are formed when the globular proteins, alpha-tubulin and beta-tubulin form dimer and undergo polymerisation. They have a diameter of about 25 nanometre.They form the structural components of flagella, cilia and centrioles. They prevent cell compression.

Your answer would be C