Question

The functions f and g are defined as follows.

f(x)= 1/(x+2)

g(x)= square root of: (x-1)

(a)
State which value of x cannot be included in the domain of f.

(b)
Find the inverse of g(x)


(c) Calculate fg(10)

Answer 1

The function f(x) cannot include x = -2 in its domain, the inverse of g(x) is g^(-1)(x) = x^2 + 1, and fg(10) is f(g(10)) = f(sqrt(9)) = f(3).

Step-by-step explanation:

(a) The function f(x) is defined as f(x) = 1/(x+2). Since the divisor (x+2) cannot equal to zero, the value of x that cannot be included in the domain of f is x = -2.

(b) To find the inverse of g(x), we need to swap the variables x and y and solve for y. So, let y = sqrt(x-1), then we have x = sqrt(y-1). Squaring both sides, we get x^2 = y-1. Rearranging, we have y = x^2 + 1. Therefore, the inverse of g(x) is g^(-1)(x) = x^2 + 1.

(c) To calculate fg(10), we substitute g(10) into f(x). We have f(g(10)) = f(sqrt(10-1)) = f(sqrt(9)) = f(3).

Answer 2

x != -2, x^2 + 1, 1/5

Step-by-step explanation:

(a) It can be understood that nothing can divide 0

thus denominator cannot be 0 meaning x cannot be -2.

(-2 + 2) = 0

(b) y = sqrt(x-1)

inverse meaning we make x the subject

square both sides

y^2 = x - 1

y^2 + 1 = x

inverse = x^2 + 1

(c) composite function:

pass g(10) as domain of f

g(10) = 3, f(3) = 1/(5) = 1/5