Question

Consider the functions f(x) = 2x and g(x) = frac{1}{x-3}" alt="\frac{1}{x-3}" align="absmiddle" class="latex-formula">≠ 3
(a) Calculate (f ° g)(4)

(b) Find $g^{-1} (x)$

(c) Write doen the domain of $g^{-1}$

Answer 1

Question 1:

For this case we must find$(f_ {0} g) (4)$knowing that:

$f (x) = 2x\\g (x) = \frac {1} {x-3}$

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

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

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

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

So, we have:

$(f_ {0} g) (x) = 2 (\frac {1} {x-3})\\(f_ {0} g) (4) = 2 (\frac {1} {4-3})\\(f_ {0} g) (4) = 1$

ANswer:

$(f_ {0} g) (4) = 1$

Question 2:

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

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

$y = \frac {1} {x-3}$

We exchange variables:

$x = \frac {1} {y-3}$

We solve for "y":

$y-3 = \frac {1} {x}\\y = \frac {1} {x} +3$

Replace "y" with $g ^ {-1} (x)$

So, we have:

$g ^ {- 1} (x) = \frac {1} {x} +3$

Answer:

$g ^ {- 1} (x) = \frac {1} {x} +3$

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:

$g ^ {- 1} (x) = \frac {1} {x} +3$

It is observed that the function is not defined for $x = 0$

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

${x | x \neq 0}$for any integer n

Answer:

All numbers, except$x = 0$

Answer 2

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,∞).