Given$ f(x+1)=2f(x)$, why does the y of the image of $y=f(x)$ decrease by half when it is translated to the left by $1$?

Question

Given $f(x+1)=2f(x)$, why does the $y$ of the image of $y=f(x)$ decrease by half when it is translated to the left by $1$?

I mean the Transformation of Graphs.

This is what I thought: Since $f(x+1)=2f(x)$, when the image $y=f(x)$ is translated to the left by $1$ and becomes $f(x+1)$ (left form), the y of the original function becomes twice as much (right form). But in fact, this is not the case.

This question seems very simple, but I don't understand it. Thank you for your answers. (Don't use the special value method)

Answer 1

You have correctly understood that, starting from an "original" graph of $f(x)$, the graph of $f(x+1)$ is the original shifted $1$ to the left, and the graph of $2f(x)$ is the original stretched vertically by a factor of $2$.

Your mistake is in interpreting the equation as "when the [graph] $y=f(x)$ is translated to the left by $1$ and becomes $f(x+1)$ (left form), the $y$ of the original function becomes twice as much (right form)". No it does not. Rather it should be : "it (the new graph) becomes twice as much as the original (right form)".

The equation says that for the graph of $f$, a shift one to the left has the same effect as stretching the graph vertically by a factor of $2$. And the graph of any $f$ satisfying the equation, e.g. $f(x) = a\cdot 2^x$ for arbitrary $a\in \mathbb R$, has this property.

And in fact that is what you see: If you shift the graph one to the left and compare it to the original graph, the: At each $x$, the $y$-value $\color{blue}{\text{of this new graph}}$ $\color{green}{\text{is}}$ twice as big as the one of the $\color{red}{\text{original } f}$.

In formula, $\color{blue}{f(x+1)} \color{green}{=} 2\color{red}{f(x)}$.

Answer 2

We have, $f(x + 1) = 2 f(x)$. In order to make it easier to understand I will change the variable. Let $x = y-1$. Now the new equation is $$f(y) = 2f(y-1)$$ $$f(y-1) = \frac{1}{2}f(y).$$ Here clearly we can see how by going 1 unit to the left our function will be decreased by a factor of $\frac{1}{2}$.

Answer 3

Take $x=7.5$. The equation tells you that $f(8.5)$ is twice as big as $f(7.5)$, which means that $f(7.5)$ is half as big as $f(8.5)$. In general, the value that $f$ takes at a point is half as big as the value it takes one unit to the right.

Let's use the same method to interpret the statement that the graph of the function $x\mapsto f(x+1)$ is the graph of the function $x\mapsto f(x)$ shifted left by one unit. The value of the function of $x\mapsto f(x+1)$ at $x=7.5$ is $f(8.5)$, that is, the same as the value of the function $x\mapsto f(x)$ at $x=8.5$. In general, the value that $x\mapsto f(x+1)$ takes at a point is the same as the value that $x\mapsto f(x)$ takes one unit to the right.