Answer:
[tex]\sqrt{-(x +72)^3} - 72 = -(3\sqrt x - 72 +72)^3[/tex]
Step-by-step explanation:
Given
[tex]g(x) = 3\sqrt x - 72[/tex]
[tex]h(x) = -(x +72)^3[/tex]
Required
Show that they are inverse functions
For g(x) and h(x) to be inverse, then:
[tex]g(h(x)) = h(g(x))[/tex]
We have:
[tex]g(x) = 3\sqrt x - 72[/tex]
Replace x with h(x)
[tex]g(h(x)) = 3\sqrt{h(x)} - 72[/tex]
Substitute value for h(x)
[tex]g(h(x)) = 3\sqrt{-(x +72)^3} - 72[/tex]
Similarly;
[tex]h(x) = -(x +72)^3[/tex]
Replace x with g(x)
[tex]h(g(x)) = -(g(x) +72)^3[/tex]
Substitute value for g(x)
[tex]h(g(x)) = -(3\sqrt x - 72 +72)^3[/tex]
Recall that:
[tex]g(h(x)) = h(g(x))[/tex]
So:
[tex]\sqrt{-(x +72)^3} - 72 = -(3\sqrt x - 72 +72)^3[/tex]
