Question

Given that function f(x) = 2/x - 8 and the composite function fg(x) = 4/x² - 2, find:

(a) g(x)
(b) g(f(x))
(c) f⁻¹(x)

a) a) g(x) = 2/x - 8; b) g(f(x)) = 4/x² - 2; c) f⁻¹(x) = 8/x - 2
b) a) g(x) = 4/x - 2; b) g(f(x)) = 2/x² - 8; c) f⁻¹(x) = 8 - 2x
c) a) g(x) = 2 - 8x; b) g(f(x)) = 4/x - 2; c) f⁻¹(x) = x/(2x - 8)
d) a) g(x) = 4 - 2/x; b) g(f(x)) = 2 - 8x; c) f⁻¹(x) = 2/x + 8

Answer 1

To find g(x), g(f(x)), and f⁻¹(x) given the functions f(x) and fg(x), we substitute the expressions and solve for each variable.

Step-by-step explanation:

(a) To find g(x), we are given the composite function fg(x) = 4/x² - 2. We can equate this to the expression for g(x) and solve for g(x). So, we have:

4/x² - 2 = g(x)

g(x) = 4/x² - 2

(b) To find g(f(x)), we substitute the expression for f(x) into g(x). So, we have:

g(f(x)) = 4/(2/x - 8)² - 2

g(f(x)) = 4/(2/x - 8)(2/x - 8) - 2

g(f(x)) = 4/(4/x² - 16/x - 16/x + 64) - 2

g(f(x)) = 4/(4/x² - 32/x + 64) - 2

g(f(x)) = 4x²/(4 - 32x + 64x²) - 2

(c) To find f⁻¹(x), we swap the roles of x and f(x) in the expression for f(x), and solve for f⁻¹(x). So, we have:

x = 2/f⁻¹(x) - 8

x + 8 = 2/f⁻¹(x)

f⁻¹(x) = 2/(x + 8)