Question

Given that f(x)=x^2+2x and g(x)=x+3, calculate(f•g)(x)=(g•f)(x)=(f•f)(x)=(g•g(x)=

Answer 1

We are given the following functions:

`\begin{gathered} f\mleft(x\mright)=x^2+2x \\ g\mleft(x\mright)=x+3 \end{gathered}`

We are asked to determine the following composition:

`(f\circ g)(x)=f(g(x))`

This means that where we have "x" in f we will replace it for the function g, like this:

`f(g(x))=(x+3)^2+2(x+3)`

Simplifying we get:

`\begin{gathered} f(g(x))=x^2+6x+9+2x+6 \\ f(g(x))=x^2+8x+15 \end{gathered}`

Now we are asked to determine the following composition:

`(g\circ f)(x)=g(f(x))`

This means that where there is "x" in g we will replace it by f:

`\begin{gathered} g(f(x))=(x^2+2x)+3 \\ g(f(x))=x^2+2x+3 \end{gathered}`

Now we are asked to determine:

`(f\circ f)(x)=f(f(x))`

Replacing the value of f in f:

`f(f(x))=(x^2+2x)^2+2(x^2+2x)`

Simplifying:

`\begin{gathered} f(f(x))=x^4+4x^3+4x^2+2x^2+4x \\ f(f(x))=x^4+4x^3+6x^2+4x \end{gathered}`

Finally, we are asked to determine the following composition:

`(g\circ g)(x)=g(g(x))`

Replacing we get:

`\begin{gathered} g(g(x))=(x+3)+3 \\ g(g(x))=x+6 \end{gathered}`