Functions f and g are defined by f:x→ax-b,where a and b are constants and g:x→x+5.Given that fg(1)=11 and gf(2)=8.

Calculate the values of a and b?

What are the values of fg(-1)and gf(3)

fg(x) = ag-b = a(x+5)-b = ax+5a-b

fg(1) = 6a-b = 11

gf(x) = f+5 = ax-b+5
gf(2) = 2a-b+5 = 8 or 2a-b = 3
So,
4a = 8
a = 2
b = 1

fg(x) = 2x+9
gf(x) = 2x+4

and so on

To calculate the values of a and b, we can use the given information.

First, let's find the value of fg(1) and gf(2) using the given functions:

fg(1) = f(g(1)) = f(1 + 5) = f(6) = 6a - b = 11

gf(2) = g(f(2)) = g(2a - b) = 2a - b + 5 = 8

Now, let's solve these equations:

6a - b = 11 (Equation 1)
2a - b + 5 = 8 (Equation 2)

We can rearrange Equation 1 to solve for b:

b = 11 - 6a

Substituting this value of b in Equation 2:

2a - (11 - 6a) + 5 = 8

Simplifying the equation:

2a - 11 + 6a + 5 = 8
8a - 6 = 8
8a = 14
a = 14/8
a = 7/4

Now substituting the value of a into b = 11 - 6a:

b = 11 - 6(7/4)
b = 44/4 - 42/4
b = 2/4
b = 1/2

Therefore, the values of a and b are a = 7/4 and b = 1/2.

Now, let's calculate fg(-1) and gf(3) using the given values of a and b:

fg(-1) = f(g(-1)) = f(-1 + 5) = f(4) = 4a - b
gf(3) = g(f(3)) = g(3a - b) = 3a - b + 5

Substituting the known values:

fg(-1) = 4(7/4) - 1/2 = 7 - 1/2 = 13/2
gf(3) = 3(7/4) - 1/2 + 5 = 21/4 - 1/2 + 20/4 = 40/4 = 10

Therefore, the value of fg(-1) is 13/2 and the value of gf(3) is 10.

To calculate the values of a and b, we need to use the given information about fg(1) and gf(2).

First, let's calculate fg(1):
fg(1) means that we apply the function f to 1, and then apply the function g to the result.
f:x→ax-b
g:x→x+5
So, fg(1) = g(f(1)).
f(1) = a*1 - b = a - b
g(f(1)) = g(a - b) = (a - b) + 5

According to the given information, we know that fg(1) = 11. So, we can write the equation:
(a - b) + 5 = 11

Next, let's calculate gf(2):
gf(2) means that we apply the function g to 2, and then apply the function f to the result.
f:x→ax-b
g:x→x+5
So, gf(2) = f(g(2)).
g(2) = 2 + 5 = 7
f(g(2)) = f(7) = a * 7 - b

According to the given information, we know that gf(2) = 8. So, we can write the equation:
a * 7 - b = 8

Now we have a system of two linear equations:
(a - b) + 5 = 11 (Equation 1)
a * 7 - b = 8 (Equation 2)

We can solve the system of equations to find the values of a and b.