Answer:
Explanation:
Given the domain and target set of functions f and g expressed as;
f(x) = 2x+3 an g(x) = 5x+7 we are to find the following;
a) f◦g
f◦g = f[g(x)]
f[g(x)] = f[5x+7]
To get f(5x+7), we will replace the variable x in f(x) 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
Hence f◦g = 10x+17
b) g◦f
g◦f = g[f(x)]
g[f(x)] = g[2x+3]
To get g(2x+3), we will replace the variable x in g(x) 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
Hence g◦f = 10x+22
c) For (f◦g)−1 (inverse of (f◦g))
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 the inverse of the function
(f◦g)−1 = (x-17)/10
d) For the function f−1◦g−1
We need to get the inverse of function f(x) and g(x) first.
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
Hence the inverse of the function
f-1(x) = (x-3)/2
For 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
Hence the inverse of the function
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-1(x)]
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
Hence g-1◦f-1 = (x-17)/10