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

###### Question:

a. give a restricted domain for f where it is invertible.

b. find the inverse of f for the domain you gave in part (a).

c. state the domain and range of the function you found in part (b).

d. verify through function composition that the function you found in part (b) is the inverse of f

## Answers

-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