Question

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

Answer 1

See Below.

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) = √(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)) &= (√(x + 8) -4)^2 + 8(√( x+ 8)-4) + 8 \\ \\ &= ((x+8) -8√(x + 8) +16) + 8√(x + 8) - 32 + 8 \\ \\ &= x + (-8√(x+8) + 8√(x + 8)) +(16 - 32 + 8 + 8) \\ \\ &= x \stackrel{\checkmark}{=} x\end{aligned}`

And:

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

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