Question

Find (i) gof and (ii) fog where
i) f(x) = x - 2, g(x) = x² + 3x + 1
ii) $f(x)=\frac{1}{x},\ \ g(x)=\frac{x-2}{x+2}$

Answer 1

i) f(x) = x - 2 and g(x) = x² + 3x + 1

gof = g(f(x)) = g(x - 2)

= (x - 2)² + 3(x - 2) + 1

= x² - 4x + 4 + 3x - 6 + 1

= x² - x - 1

hence, gof = x² - x - 1

fog = f(g(x)) = f(x² + 3x + 1)

= (x² + 3x + 1) - 2

= x² + 3x - 1

hence, fog = x² + 3x - 1


ii) f(x) = 1/x and g(x) = (x - 2)/(x + 2)

gof = g(f(x))

= g(1/x)

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

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

hence, gof = (1 - 2x)/(1 + 2x)


fog = f(g(x))

= f[(x - 2)/(x + 2)]

= 1/[(x - 2)/(x + 2)]

= (x + 2)/(x - 2)

hence, fog = (x + 2)/(x - 2)