Algebraic Operations on Functions

Given $f\left(x+1\right)=3x+5$. Evaluate

$$\begin{gathered} a. f\left(-2\right) \\ b. f\left(2x\right) \end{gathered}$$

If $2f\left(x\right)+3f(-x)=2x+1$, then find $f\left(x\right)$.

$f\left(x\right)=x^2+4x+6$ and $g\left(x\right)=x^2+2x+8$. Prove that addition of functions is commutative using given functions.

$f\left(x\right)=2x^2+4x+5$ and $g\left(x\right)=x^2+3x+4$. Prove that addition of functions is commutative using given functions.

$f\left(x\right)=x^2+2x+2$ and $g\left(x\right)=x+1$. Prove that addition of functions is commutative using given functions.

$f\left(x\right)=x^2+4x+2$ and $g\left(x\right)=x-1$. Prove that addition of functions is commutative using given functions.

$f\left(x\right)=x^2+3$ and $g\left(x\right)=x+4$. Prove that addition of functions is commutative using given functions.

If $f\left(x\right)=x^2$ and $g\left(x\right)=2x-1$, then find the value of $\left(f-g\right)\left(41\right)$.

If $f\left(x\right)=x^2$ and $g\left(x\right)=2x+1$, then find the value of $\left(f+g\right)\left(59\right)$.

Let $f\left(x\right)=a{x}^2+bx+c \text{and} g\left(x\right)=p{x}^2+qx$ with $g\left(1\right)=f\left(1\right), g\left(2\right)-f\left(2\right)=1, g\left(3\right)-f\left(3\right)=4,$ then $g\left(4\right)-f\left(4\right)$is: