robert7248
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Consider the functions f(x) = 2x and g(x) = [tex]\frac{1}{x-3}[/tex]≠ 3

(a) Calculate (f ° g)(4)

(b) Find [tex]g^{-1} (x)[/tex]

(c) Write doen the domain of [tex]g^{-1}[/tex]

Consider the functions fx 2x and gx texfrac1x3tex 3 a Calculate f g4 b Find texg1 xtex c Write doen the domain of texg1tex class=

Respuesta :

carlosego

Question 1:

For this case we must find[tex](f_ {0} g) (4)[/tex]knowing that:

[tex]f (x) = 2x\\g (x) = \frac {1} {x-3}[/tex]

By definition we have that, be two functions f (x) and g (x), then the composite function of f with g is:

[tex](g_ {0} f) (x) = g [f (x)][/tex]

In this case, they ask us for the function composed of g with f:

[tex](f_ {0} g) (x) = f [g (x)][/tex]

So, we have:

[tex](f_ {0} g) (x) = 2 (\frac {1} {x-3})\\(f_ {0} g) (4) = 2 (\frac {1} {4-3})\\(f_ {0} g) (4) = 1[/tex]

ANswer:

[tex](f_ {0} g) (4) = 1[/tex]

Question 2:

For this case, we must find the inverse function of [tex]g (x) = \frac {1} {x-3}[/tex], given by: [tex]g ^ {- 1} (x)[/tex]

To do this, replace g(x) with y:

[tex]y = \frac {1} {x-3}[/tex]

We exchange variables:

[tex]x = \frac {1} {y-3}[/tex]

We solve for "y":

[tex]y-3 = \frac {1} {x}\\y = \frac {1} {x} +3[/tex]

Replace "y" with [tex]g ^ {-1} (x)[/tex]

So, we have:

[tex]g ^ {- 1} (x) = \frac {1} {x} +3[/tex]

Answer:

[tex]g ^ {- 1} (x) = \frac {1} {x} +3[/tex]

Question 3:

For this case, we have by definition, the domain of a function f (x), is the set of all the values ​​for which the function is defined.

We must find the domain of the following function:

[tex]g ^ {- 1} (x) = \frac {1} {x} +3[/tex]

It is observed that the function is not defined for [tex]x = 0[/tex]

Then the domain is given by all the values ​​of x, except 0.

[tex]{x | x \neq 0}[/tex]for any integer n

Answer:

All numbers, except[tex]x = 0[/tex]

zainebamir540

Answer to Q1:

(fog)(4) =  2

Step-by-step explanation:

We have given two function. We have to find their composition.

f(x) = 2x    and g(x)= 1 / x-3

(fog)(x) = ?   and (fog)(4) = ?

The formula to find composition is:

(fog)(x) = f(g(x))

(fog)(x) = f(1 / x-3)

(fog)(x) =  2(1 / x-3)

(fog)(x) =  2 / x-3

Putting x = 4 in above equation, we have

(fog)(4) = 2 / 4-3

(fog)(4) =  2 / 1

(fog)(4) =  2

Answer to Q2:

g⁻¹(x) = 1/x+3

Step-by-step explanation:

We have given a function and we have to find its inverse.

g(x) = 1 / x-3    

g⁻¹(x) = ?

Let y = g(x)

y = 1 / x-3

We have to separate x from above equation.

y(x-3) = 1

x-3 = 1 / y

Adding 3 to both sides of above equation, we have

x-3+3 = 1/y+3

x = 1/y+3

Putting x = g⁻¹(y) in above equation, we have

g⁻¹(y) = 1/y+3

Replacing y with x , we have

g⁻¹(x) = 1/x+3 which is the answer.

Answer to Q3:

(-∞,0)∪(0,∞)

Step-by-step explanation:

Since   g⁻¹(x) = 1/x+3

We have to find the domain of above function.

Domain is defined as the set of values of independent variable where function is defined.

Hence given function contain 1/x term which is  defined all real values except at x = 0.

The term 3 is defined at all real values.

Hence,g⁻¹(x) has domain equal to all real values except x = 0.

dom g⁻¹(x) = (-∞,0)∪(0,∞).

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