4. Dina has a calculating machine, labelled $f$, that takes one number as input and calculates an output. The machine $f$ calculates its output by multiplying its input by 2 and then subtracting 3. For example, if Dina inputs 2.16 into $f$, the output is 1.32. If Dina inputs a number $x$ into $f$, she gets a first output which she then inputs back into $f$ to obtain a second output, which is -35 . What is the value of $x$ ?
4. Solution 1
Dina's machine works by taking an input, multiplying by 2 and then subtracting 3 .
If we have the output and want to obtain the input, we must reverse these operations and so we take the output, add 3 and then divide by 2 .
Starting with the second output -35 , we obtain $-35+3=-32$ and then $\frac{-32}{2}=-16$, which was the second input.
Since the second input was the first output, then the first output was -16 .
Starting with the first output -16 , we obtain $-16+3=-13$ and then $\frac{-13}{2}$, which was the first input.
Therefore, the first input was $-\frac{13}{2}$ or -6.5 .
Solution 2
Starting with the first input $x$, Dina's machine multiplies it by 2 to obtain $2 x$ and then subtracts 3 to obtain $2 x-3$.
Dina inputs this output back into the machine.
The machine multiplies the input by 2 to obtain $2(2 x-3)=4 x-6$ and then subtracts 3 to obtain $4 x-9$.
Since the second output is -35 , then $4 x-9=-35$ or $4 x=-26$ and so $x=\frac{-26}{4}=-\frac{13}{2}$ or -6.5 .
ANSWER: $-\frac{13}{2}$