Chapter 3.5, Problem 52HP

To determine

If $af(x)$ and $f(ax)$ are equivalent functions.

Answer to Problem 52HP

No, both given functions are not equivalent. So, Alex is not correct.

Explanation of Solution

Given information: Two functions $af(x)$ and $f(ax)$ to check if these are equivalent.

Formula used: Any function is written in form $y=f(x)$ . So, $af(x)$ means that whole function is being multiplied by constant a and $f(ax)$ means each x value is being multiplied by a.

Calculation: To check if both are equivalent functions, let us consider a function,

  $y=f(x)=2x+1$

Then, $af\left(x\right)=a\left(2x+1\right)=2ax+a$

And accordingly, on replacing each x with ax,

  $f\left(ax\right)=2\left(ax\right)+1=2ax+1$

Clearly, $af\left(x\right)\ne f(ax)$ .

Conclusion: So, both functions are surely not equivalent, so, Alex is not correct.