Question

We can express the functions in Exercise 3 algebraically as

$\begin{array}{c}f\left(x\right)=\_\_\_\_\_\\ g\left(x\right)=\_\_\_\_\_\\ \left(f\circ g\right)\left(x\right)=\_\_\_\_\_\\ \left(g\circ f\right)\left(x\right)=\_\_\_\_\_\end{array}$

Short answer

The algebraic forms of the required functions are:

$\begin{array}{c}f\left(x\right)=x+1\\ g\left(x\right)=2x\\ \left(f\circ g\right)\left(x\right)=2x+1\\ \left(g\circ f\right)\left(x\right)=2x+2\end{array}$

Step-by-step solution

– Composition of function.

Composition of functions is defined as:

$\begin{array}{l}\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)\\ \left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)\end{array}$

– Given information.

Function is f “add one” and the rule of the function g is “multiply by 2 ”.

–Algebraic form of given functions.

The given functions can be written as:

$\begin{array}{c}f\left(x\right)=x+1\\ g\left(x\right)=2x\end{array}$

– Algebraic forms of composition of functions.

$\begin{array}{c}\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)\\ =f\left(2x\right)\\ =\left(2x\right)+1\\ =2x+1\end{array}$

And,

$\begin{array}{c}\left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)\\ =g\left(x+1\right)\\ =2\left(x+1\right)\\ =2x+2\end{array}$

Thus, the algebraic forms of the required functions are:

$\begin{array}{c}f\left(x\right)=x+1\\ g\left(x\right)=2x\\ \left(f\circ g\right)\left(x\right)=2x+1\\ \left(g\circ f\right)\left(x\right)=2x+2\end{array}$