Question

Show algebraically that f and g are inverse functions:
g = sqrt(x+8)-4
f=x^2+8x+8

Answer 1

Answer:

See Below.

Step-by-step explanation:

By definition, if two functions, f and g, are inverses, then:

$\displaystyle f(g(x)) = g(f(x)) = x$

We are given the two functions:

$\displaystyle g(x) = \sqrt{x + 8} -4 \text{ and } f(x) = x^2 + 8x + 8$

Find f(g(x)) and g(f(x)):

$\displaystyle \begin{aligned} f(g(x)) &= (\sqrt{x + 8} -4)^2 + 8(\sqrt{ x+ 8}-4) + 8 \\ \\ &= ((x+8) -8\sqrt{x + 8} +16) + 8\sqrt{x + 8} - 32 + 8 \\ \\ &= x + (-8\sqrt{x+8} + 8\sqrt{x + 8}) +(16 - 32 + 8 + 8) \\ \\ &= x \stackrel{\checkmark}{=} x\end{aligned}$

And:

$\displaystyle \begin{aligned}g(f(x)) &= \sqrt{(x^2 + 8x + 8) + 8} - 4 \\ \\ &= \sqrt{x^2 + 8x+16 } - 4 \\ \\ &=\sqrt{(x+ 4)^2} - 4 \\ \\ &= (x +4) - 4 \\ \\ &= x \stackrel{\checkmark}{=} x \end{aligned}$

Hence, f and g are indeed inverses of each other.