Question

F(x) = x² + 6x-2, g(x) = x-6, find f(g(4)) and (gof)(-4).

Answer 1

f(g(4)) = -10

(g o f)(-4) = -16

Explanation:

Function composition is an operation where one or more functions are substituted into another function.

Given functions:

`\begin{cases}f(x)=x^2+6x-2\\g(x)=x-6\end{cases}`

The given composite function f(g(4)) means to substitute the function g(4) in place of the x in function f(x). The function g(4) is the value of function g(x) when x = 4.

`\begin{aligned}f\left(g(4)\right)&=\left(g(4)\right)^2+6\left(g(4)\right)-2\\&=\left(4-6\right)^2+6\left(4-6\right)-2\\&=\left(-2\right)^2+6\left(-2\right)-2\\&=4-12-2\\&=-8-2\\&=-10\end{aligned}`

The given composite function (g o f)(-4) means to substitute the function f(x) in place of the x in function g(x), then substitute x = -4 into the resulting function.

`\begin{aligned}\left(g \circ f \right)(-4)&=g\left(f(-4)\right)\\&=\left(f(-4)\right)-6\\&=((-4)^2+6(-4)-2)-6\\&=(16-24-2)-6\\&=-10-6\\&=-16\end{aligned}`