Question

The domain and target set of functions f and g isR. The functions are definedas:(b)•f(x) = 2x+ 3•g(x) = 5x+ 7(a)f◦g?(b)g◦f?(c) (f◦g)−1?(d)f−1◦g−1?(e)g−1◦f−1?

Answer 1

Step-by-step explanation:

The domain and target set of functions f and g given is expressed as;

f(x) = 2x+3 an g(x) = 5x+7 on R. To calculate the given functions, the following steps must be followed.

a) f◦g

f◦g = f(g(x)]) = f(5x+7)

To solve for the function f(5x+7), the variable x in f(x) will be replaced with 5x+7 as shown;

f(x) = 2x+3

f(5x+7) = 2(5x+7)+3

f(5x+7) = 10x+14+3

f(5x+7) = 10x+17

Therefore the function f◦g is equivalent to 10x+17

b) For the composite function g◦f  

g◦f = g(f(x)])

g(f(x)) = g(2x+3))

To drive the functon g(2x+3), the variable x in g(x) will be replaced with 2x+3 as shown;

g(x) = 5x+7

g(2x+3) = 5(2x+3)+7

g(2x+3) = 10x+15+7

g(2x+3) = 10x+22

This shoes that the composite function g◦f = 10x+22

c) To get the inverse of the composite function f◦g i.e (f◦g)⁻¹

Given (f◦g) = 10x+17

To find the inverse, first we will replace (f◦g) with variable y to have;

y = 10x+17

Then we will interchange variable y for x:

x = 10y+17

We will then make y the subject of the formula;

10y = x-17

y = (x-17)/10

Hence (f◦g)⁻¹ = (x-17)/10

d) For the function f⁻¹◦g⁻¹

First we need to calculate for the inverse of function f(x) and g(x) as shown:

For f⁻¹(x):

Given f(x)= 2x+3

To find the inverse, first we will replace f(x) with variable y to have;

y = 2x+3

Then we will interchange variable y for x:

x = 2y+3

We will then make y the subject of the formula;

2y = x-3

y = (x-3)/2

f⁻¹(x) = (x-3)/2

Similarly for the function g⁻¹(x):

Given g(x)= 5x+7

To find the inverse, first we will replace g(x) with variable y to have;

y = 5x+7

Then we will interchange variable y for x:

x = 5y+7

We will then make y the subject of the formula;

5y = x-7

y = (x-7)/5

g⁻¹(x) = (x-7)/5

Now to get f⁻¹◦g⁻¹

f⁻¹◦g⁻¹= f⁻¹(g⁻¹(x))

f⁻¹(g⁻¹(x)) = f⁻¹((x-7)/5)

Since f⁻¹(x) = (x-3)/2

f⁻¹((x-7)/5) = [(x-7)/5)-3]/2

= [(x-7)-15/5]/2

= [(x-7-15)/5]/2

= [x-22/5]/2

= (x-22)/10

Hence f⁻¹◦g⁻¹= (x-22)/10

e) For the composite function g⁻¹◦f⁻¹

g⁻¹◦f⁻¹= g⁻¹[f⁻¹x)]

g⁻¹[f⁻¹(x)] = g⁻¹((x-3)/2)

Since g⁻¹(x) = (x-7)/5

g⁻¹(x-3/2) = [(x-3/2)-7]/5

= [(x-3)-14)/2]/5  

= [(x-17)/2]/5

= (x-17)/10

Therefore the composite function g⁻¹◦f⁻¹= (x-17)/10