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
