Chapter 1.4, Problem 47E

Solution

To find: Which statement is not true.

The statement $f\left(g\right)\left(x\right)=g\left(f\right)\left(x\right)$ is not true.

Given information:

Suppose $f$ and $g$ are functions with domain all real numbers.

Calculation:

Suppose $f\left(x\right)=x$ and $g\left(x\right)=x^2$ are functions with domain all real numbers.

Option (A),

  $\left(f+g\right)\left(x\right)=\left(g+f\right)\left(x\right)$

Rewrite the above equation:

  $f\left(x\right)+g\left(x\right)=g\left(x\right)+f\left(x\right)$

Substitute all values and check it is equal or not.

  $\begin{array}{l}x+x^2=x^2+x\\ x+x^2=x+x^2\end{array}$

Therefore, the statement $\left(f+g\right)\left(x\right)=\left(g+f\right)\left(x\right)$ is true.

Option (B),

Suppose $f\left(x\right)=x$ and $g\left(x\right)=x^2$ are functions with domain all real numbers.

  $\left(fg\right)\left(x\right)=\left(gf\right)\left(x\right)$

Rewrite the above equation:

  $f\left(x\right)g\left(x\right)=g\left(x\right)f\left(x\right)$

Substitute all values and check it is equal or not.

  $\begin{array}{c}x\times x^2=x^2\times x\\ x^3=x^3\end{array}$

Therefore, the statement $\left(fg\right)\left(x\right)=\left(gf\right)\left(x\right)$ is true.

Option (C),

Suppose $f\left(x\right)=x-1$ and $g\left(x\right)=x^2$ are functions with domain all real numbers.

  $f\left(g\left(x\right)\right)=g\left(f\left(x\right)\right)$

Substitute all values and check it is equal or not.

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

Therefore, the statement $\left(fg\right)\left(x\right)=\left(gf\right)\left(x\right)$ is not true.

Option (D),

Suppose $f\left(x\right)=x$ and $g\left(x\right)=x^2$ are functions with domain all real numbers.

  $\left(f-g\right)\left(x\right)=-\left(g-f\right)\left(x\right)$

Rewrite the above equation:

  $f\left(x\right)-g\left(x\right)=-\left(g\left(x\right)-f\left(x\right)\right)$

Substitute all values and check it is equal or not.

  $\begin{array}{c}x-x^2=-\left(x^2-x\right)\\ x-x^2=x-x^2\end{array}$

Therefore, the statement $\left(f-g\right)\left(x\right)=-\left(g-f\right)\left(x\right)$ is true.

Option (E),

Suppose $f\left(x\right)=x$ and $g\left(x\right)=x^2$ are functions with domain all real numbers.

  $\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$

Rewrite the above equation:

  $f\left(g\left(x\right)\right)=f\left(g\left(x\right)\right)$

Substitute all values and check it is equal or not.

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

Therefore, the statement $\left(f-g\right)\left(x\right)=-\left(g-f\right)\left(x\right)$ is true.