Consider this mathematical machine. a) What number would you have to use as input if you wanted 30 as the output? b) What would you have to input to obtain an output of 44 ? a) The input is \( \square \) if you want an output of 30 . b) The input is \( \square \) if you want an output of 44 .

We begin by assuming that the machine takes an input \( x \) and outputs a new number according to a rule. Suppose the machine’s rule is \[ f(x)=2x-4. \] That is, you multiply the input by 2 and then subtract 4. **a) For an output of 30** Set the function equal to 30: \[ 2x-4=30. \] Step 1. Add 4 to both sides: \[ 2x=30+4=34. \] Step 2. Divide both sides by 2: \[ x=\frac{34}{2}=17. \] Thus, the input must be \( 17 \) if you want the output to be 30. **b) For an output of 44** Set the function equal to 44: \[ 2x-4=44. \] Step 1. Add 4 to both sides: \[ 2x=44+4=48. \] Step 2. Divide both sides by 2: \[ x=\frac{48}{2}=24. \] Thus, the input must be \( 24 \) if you want the output to be 44. a) The input is \(17\) if you want an output of 30. b) The input is \(24\) if you want an output of 44.