12 multiple-choice questions, progressively harder.
Which operation undoes "add 8"?
Solution
Correct answer: A
Adding and subtracting are opposite operations, so the undo of "add 8" reverses it.
(x+8)−8=x(x + 8) - 8 = x(x+8)−8=x
Subtracting 888 brings every output back to the input it came from.
Which operation undoes "multiply by 4"?
Correct answer: C
Multiplying and dividing are opposite operations, so "multiply by 4" is undone by dividing by 444.
4x4=x\frac{4x}{4} = x44x=x
If fff sends 222 to 777, what does f−1f^{-1}f−1 send 777 to?
Correct answer: B
The inverse reverses each input-output pair, so (2,7)(2, 7)(2,7) becomes (7,2)(7, 2)(7,2).
f(2)=7 ⟹ f−1(7)=2f(2) = 7 \;\Longrightarrow\; f^{-1}(7) = 2f(2)=7⟹f−1(7)=2
The rule "subtract 5" is undone by which rule?
Correct answer: D
Subtracting is reversed by adding the same amount.
(x−5)+5=x(x - 5) + 5 = x(x−5)+5=x
If f(x)=x+10f(x) = x + 10f(x)=x+10, then f−1(x)=f^{-1}(x) = {}f−1(x)=?
Undo "add 10" by subtracting 101010.
f−1(x)=x−10f^{-1}(x) = x - 10f−1(x)=x−10
The reciprocal 1x+10\dfrac{1}{x + 10}x+101 is a different object; it does not undo fff.
Which operation undoes "divide by 6"?
Dividing is reversed by multiplying by the same number.
x6⋅6=x\frac{x}{6} \cdot 6 = x6x⋅6=x
For f(x)=x−7f(x) = x - 7f(x)=x−7, find f−1(x)f^{-1}(x)f−1(x).
Undo "subtract 7" by adding 777.
f−1(x)=x+7f^{-1}(x) = x + 7f−1(x)=x+7
A table gives f(1)=3f(1) = 3f(1)=3, f(2)=6f(2) = 6f(2)=6, and f(4)=8f(4) = 8f(4)=8. What is f−1(6)f^{-1}(6)f−1(6)?
Look for the input whose output is 666, then swap the pair.
(2,6) → (6,2)(2, 6) \;\to\; (6, 2)(2,6)→(6,2)
So f−1(6)=2f^{-1}(6) = 2f−1(6)=2.
The inverse of a function reverses each pair (a,b)(a, b)(a,b) into which pair?
Taking an inverse swaps the input and output of every pair.
(a,b) → (b,a)(a, b) \;\to\; (b, a)(a,b)→(b,a)
If f−1(x)=x−4f^{-1}(x) = x - 4f−1(x)=x−4, what is f(x)f(x)f(x)?
A function and its inverse undo each other, so fff reverses "subtract 4."
f(x)=x+4f(x) = x + 4f(x)=x+4
Check: f−1(f(x))=(x+4)−4=xf^{-1}(f(x)) = (x + 4) - 4 = xf−1(f(x))=(x+4)−4=x.
For f(x)=x+2f(x) = x + 2f(x)=x+2, evaluate f−1(10)f^{-1}(10)f−1(10).
The inverse is f−1(x)=x−2f^{-1}(x) = x - 2f−1(x)=x−2, so evaluate it at 101010.
f−1(10)=10−2=8f^{-1}(10) = 10 - 2 = 8f−1(10)=10−2=8
Check: f(8)=8+2=10f(8) = 8 + 2 = 10f(8)=8+2=10.
If f(x)=4xf(x) = 4xf(x)=4x, then f−1(x)=f^{-1}(x) = {}f−1(x)=?
Undo "multiply by 4" by dividing by 444.
f−1(x)=x4f^{-1}(x) = \frac{x}{4}f−1(x)=4x
The reciprocal 14x\dfrac{1}{4x}4x1 is a common trap; it does not undo fff.
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