Question

Manish writes the functions g(x) = rootindex 3 startroot negative x endroot â€"" 72 and h(x) = â€""(x 72)3. which pair of expressions could manish use to show that g(x) and h(x) are inverse functions? rootindex 3 startroot x cubed endroot minus 72 and negative (rootindex 3 startroot negative x endroot 72) cubed rootindex 3 startroot negative x cubed endroot minus 72 and negative (rootindex 3 startroot negative x endroot 72) cubed rootindex 3 startroot (x 72) cubed endroot minus 72 and negative (rootindex 3 startroot negative x endroot minus 72 72) cubed rootindex 3 startroot (negative x 72) cubed endroot minus 72 and negative (rootindex 3 startroot negative x endroot minus 72 72) cubed

Answer 1

Here we want to find the expressions we need to use to see if the functions g(x) and h(x) are inverses of each other.

Two functions f(x) and g(x) are inverses if:

f( g(x) ) = x

g( f(x) ) = x

In this case, we have the functions:

g(x) = ∛(-x) - 72

h(x) =  -(x + 72)^3

Then the expressions we need to check are:

g( h(x) ) = ∛(-h(x)) - 72 = ∛(+(x + 72)^3) - 72 = (x + 72) - 72 = x

h( g(x) ) = -(g(x) + 72)^3 = -(∛(-x) - 72 + 72)^3 = -(∛(-x) )^3 = x

So we found that the two expressions needed are:

∛((x + 72)^3) - 72  and -(∛(-x) - 72 + 72)^3

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