function $f$ has the properties that $f(1) = 6$

Question

I am not sure what exactly the question wants me to do:

The function $f$ has the properties that $f(1) = 6$ and $f(2x + 1) = 3f(x)$ for every integer $x$. What is the value of $f(63)$?

What exactly does it mean when $f(1) = 6$? and what is its relation with the other functions?

Answer 1

$$f(1)=6$$ and $$f(2x+1)=3f(x)$$ so
$$f(2(1)+1)=f(3)=3f(1)=3\cdot 6=18$$ $$f(2(3)+1)=f(7)=3f(3)=3\cdot 18=54 $$ $$f(2(7)+1)=f(15)=3f(7)=3\cdot 54=162 $$ $$f(2(15)+1)=f(31)=3f(15)=3\cdot 162=486$$ $$f(2(31)+1)=f(63)=3f(31)=3\cdot 486=1458 $$

Answer 2

A function is a way of assigning outputs to inputs. When someone says $f(1)=6$, they are saying that the value of $f$ is 6 when the input is 1.

Then, the equation $f(2x+1)=3f(x)$ is saying that two values of $f$ are related. Which two? The values of any pair of inputs $x$ and $2x+1$.

For instance, if $x=1$, then $2x+1=3$. Since we know $f(1)=6$, the equation above is telling us $$f(3) = f(2\cdot 1 + 1)=3f(1)=18$$

Now that you know the value of the function when the input is 3, you can repeat this idea for $x=3$ to figure out different values of the function.