Question

Given two functions f(x)=10-2x,xER and g(x)=x+6/2,xER.

i)find (f•g)(x)
ii)evaluate (f•g)(8)
iii)find f^-1(x)
iv)find(g•f^-1)(x)​

Answer 1

i) To find (f•g)(x), we need to perform the composition of functions f and g. The composition is denoted by (f•g)(x) and is calculated by substituting the expression for g(x) into f(x).

(f•g)(x) = f(g(x)) = f((x+6)/2) = 10 - 2 * ((x+6)/2) = 10 - (x + 6) = 4 - x

ii) To evaluate (f•g)(8), we substitute x = 8 into the expression we found in part i.

(f•g)(8) = 4 - 8 = -4

iii) To find the inverse function of f(x), we swap the roles of x and f(x) and then solve for x.

f(x) = 10 - 2x

y = 10 - 2x

x = 10 - 2y

2y = 10 - x

y = (10 - x)/2

So, f^-1(x) = (10 - x)/2

iv) To find (g•f^-1)(x), we perform the composition of functions g and f^-1.

(g•f^-1)(x) = g(f^-1(x)) = g((10 - x)/2) = ((10 - x)/2) + 6

Therefore, (g•f^-1)(x) = (10 - x)/2 + 6.

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