Chapter 2, Problem 9PS

(a)

To determine

To calculate: The composition of the function $\left(f\circ g\right)\left(x\right)$ if the functions are defined as $f\left(x\right)=4x\text{ and }g\left(x\right)=x+6$.

(b)

To determine

To calculate: The composition of the function ${\left(f\circ g\right)}^{-1}\left(x\right)$ if the functions are defined as $f\left(x\right)=4x\text{ and }g\left(x\right)=x+6$.

(c)

To determine

To calculate: The function $f^{-1}\left(x\right)\text{ and }{g}^{-1}\left(x\right)$ if the functions are defined as $f\left(x\right)=4x\text{ and }g\left(x\right)=x+6$.

(d)

To determine

To calculate: The function $\left(g^{-1}\circ f^{-1}\right)\left(x\right)$ if the functions are defined as $f\left(x\right)=4x\text{ and }g\left(x\right)=x+6$ and also compare the result with part (b).

(e)

To determine

To calculate: The values from part (a) to (d) if the functions are defined as $f\left(x\right)=x^3+1\text{ and }g\left(x\right)=2x$.

(f)

To determine

To calculate: The values from part (a) to (d) if the one-to-one function is defined as $f\left(x\right)=\sqrt{x}\text{ and }g\left(x\right)=x^2$.

(g)

To determine

The conjecture about ${\left(f\circ g\right)}^{-1}\left(x\right)\text{ and }\left(g^{-1}\circ f^{-1}\right)\left(x\right)$.