If f:N→N, g:N→N and h:N→R is defined f(x)=3x-5, g(y)=6y² and h(z)=tan⁡z, find ho(gof)

The correct option is tan(6(3x-5)²).

To find the composite function, we need to understand the concept of function composition. The composition of functions involves applying one function to the result of another function. Specifically, for three functions $f$, $g$, and $h$, the composite function is evaluated from the innermost function to the outermost function.

This composition is defined as:

$(h\circ (g\circ f\left)\right)\left(x\right)=h\left(g\right(f\left(x\right)\left)\right)$

Let's break this down step-by-step:

Step 1: Identify the given functions

The innermost function is defined as:

$f\left(x\right)=3x-5$

The middle function is defined as:

$g\left(y\right)=6y^2$

The outermost function is defined as:

$h\left(z\right)=\tan \left(z\right)$

Step 2: Find the composition of the first two functions

By definition, the composite function of $g$ and $f$ is:

$(g\circ f)\left(x\right)=g\left(f\right(x\left)\right)$

This means we substitute the entire expression of $f\left(x\right)$ in place of the variable $y$ in the function $g\left(y\right)$:

$g\left(f\right(x\left)\right)=6{(3x-5)}^2$

Step 3: Find the final composition

Next, we apply the outermost function $h$ to the result we obtained in Step 2:

$(h\circ (g\circ f\left)\right)\left(x\right)=h\left(g\right(f\left(x\right)\left)\right)$

We substitute the expression $6{(3x-5)}^2$ in place of the variable $z$ in the function $h\left(z\right)=\tan \left(z\right)$:

$h\left(g\right(f\left(x\right)\left)\right)=\tan \left(6{(3x-5)}^2\right)$

Therefore, the final composite function is:

$\tan \left(6{(3x-5)}^2\right)$