Chapter 1.5, Problem 97E

To determine

To find: the error in finding $\left(fog\right)(x)$

Explanation of Solution

Given information: $f(x)=2x+1,g(x)=3x+2$

According to the given calculation,

  $\left(fog\right)(x)=3(2x+1)+2=6x+5$ (i)

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

Put $g(x)=3x+2$

  $\left(fog\right)(x)=f(g(x))=f(3x+2)$

So, according to this, replace $x$ in $f(x)=2x+1$ with $3x+2$

Hence,

  $\left(fog\right)(x)=f(3x+2)=2(3x+2)+1$ (ii)

From (i) and (ii),

It can be observed that there is an error in step $\left(fog\right)(x)=3(2x+1)+2$ (see (i))

The reason behind this error is that instead of replacing $x$ in $f(x)=2x+1$ with $g(x)=3x+2$ ,

  $x$ in $g(x)=3x+2$ is replaced with $f(x)=2x+1$

That is instead of finding $\left(fog\right)(x)$ , $\left(gof\right)(x)$ has been computed.

Let’s find $\left(gof\right)(x)$ to check this thing

  $\left(gof\right)(x)=g(f(x))=g(2x+1)=3(2x+1)+2=6x+3+2=6x+5$ (iii)

From (i) and (iii),

It can be seen that instead of finding $\left(fog\right)(x)$ , $\left(gof\right)(x)$ has been computed.