Question

Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x. (5 points)

f(x) = quantity x minus nine divided by quantity x plus five. and g(x) = quantity negative five x minus nine divided by quantity x minus one.

Answer 1

the reason that these are inverses is because if you switch x with z

and g(f(x))=x, then (x-9)/(x+5) is (z-9)/(z+5)=x. which implies that:

(z+5) - 14/(z+5)=1-(14/(z-5))=x.

(1/(1 - x) * 14) + 5 = z = g(x) maybe i messed up?

Answer 2

Explanation:

f(x) = (x − 9) / (x + 5)

g(x) = (-5x − 9) / (x − 1)

To find f(g(x)), substitute g(x) into f(x).

f(g(x)) = [(-5x − 9) / (x − 1) − 9] / [(-5x − 9) / (x − 1) + 5]

Multiply top and bottom by x − 1.

f(g(x)) = [(-5x − 9) − 9(x − 1)] / [(-5x − 9) + 5(x − 1)]

Simplify.

f(g(x)) = (-5x − 9 − 9x + 9) / (-5x − 9 + 5x − 5)

f(g(x)) = (-14x) / (-14)

f(g(x)) = x

To find g(f(x)), substitute f(x) into g(x).

g(f(x) = [-5(x − 9) / (x + 5) − 9] / [(x − 9) / (x + 5) − 1]

Multiply top and bottom by x + 5.

g(f(x) = [-5(x − 9) − 9(x + 5)] / [(x − 9) − (x + 5)]

Simplify.

g(f(x) = (-5x + 45 − 9x − 45) / (x − 9 − x − 5)

g(f(x) = (-14x) / (-14)

g(f(x) = x