12 multiple-choice questions, progressively harder.
The notation f−1f^{-1}f−1 stands for which of these?
Solution
Correct answer: A
The symbol f−1f^{-1}f−1 names the inverse function for composition, the rule that reverses fff.
f−1(f(x))=xf^{-1}(f(x)) = xf−1(f(x))=x
It is not the reciprocal 1f\frac{1}{f}f1, a derivative, or a square.
Which function is the inverse of f(x)=x+5f(x) = x + 5f(x)=x+5?
Correct answer: D
Undo 'add 555' by 'subtract 555'. Solve y=x+5y = x + 5y=x+5 for xxx.
x=y−5 ⇒ f−1(x)=x−5x = y - 5 \;\Rightarrow\; f^{-1}(x) = x - 5x=y−5⇒f−1(x)=x−5
The reciprocal 1x+5\frac{1}{x+5}x+51 is not the inverse.
If f(3)=7f(3) = 7f(3)=7 and fff is one-to-one, what is f−1(7)f^{-1}(7)f−1(7)?
Since f(3)=7f(3) = 7f(3)=7, the inverse sends 777 back to 333.
f−1(7)=3f^{-1}(7) = 3f−1(7)=3
The inverse reverses each input-output pair.
Which function is the inverse of f(x)=2xf(x) = 2xf(x)=2x?
Correct answer: C
Undo 'times 222' by 'divide by 222'. Solve y=2xy = 2xy=2x.
x=y2 ⇒ f−1(x)=x2x = \frac{y}{2} \;\Rightarrow\; f^{-1}(x) = \frac{x}{2}x=2y⇒f−1(x)=2x
The reciprocal 12x\frac{1}{2x}2x1 is a different function.
For a function fff with an inverse, domf−1\operatorname{dom} f^{-1}domf−1 equals which set?
The inverse takes outputs of fff back to inputs, so its inputs are exactly the values fff produces.
domf−1=rangef\operatorname{dom} f^{-1} = \operatorname{range} fdomf−1=rangef
Domain and range trade places when you invert.
Which function is the inverse of f(x)=x−4f(x) = x - 4f(x)=x−4?
Correct answer: B
Undo 'subtract 444' by 'add 444'. Solve y=x−4y = x - 4y=x−4.
x=y+4 ⇒ f−1(x)=x+4x = y + 4 \;\Rightarrow\; f^{-1}(x) = x + 4x=y+4⇒f−1(x)=x+4
The choice 4−x4 - x4−x negates as well as shifts, so it is not the inverse.
Which of these functions is one-to-one on all real numbers?
A function is one-to-one when different inputs give different outputs. Test each rule.
(−1)2=12,∣−1∣=∣1∣,buta3=b3⇒a=b(-1)^2 = 1^2, \quad |-1| = |1|, \quad \text{but} \quad a^3 = b^3 \Rightarrow a = b(−1)2=12,∣−1∣=∣1∣,buta3=b3⇒a=b
Squaring, absolute value, and the constant all repeat outputs; only x3x^3x3 is one-to-one.
f−1f^{-1}f−1 means the inverse of fff. Which statement is true?
The superscript in f−1f^{-1}f−1 marks the compositional inverse, not the reciprocal.
f−1≠1fin generalf^{-1} \ne \frac{1}{f} \quad \text{in general}f−1=f1in general
For f(x)=x+3f(x) = x + 3f(x)=x+3, the inverse is x−3x - 3x−3, while 1x+3\frac{1}{x+3}x+31 is a different function.
Taking its codomain to be its range, a function has an inverse exactly when it is:
With the codomain fixed as the range, invertibility and one-to-oneness are the same condition.
f(a)=f(b)⇒a=bf(a) = f(b) \Rightarrow a = bf(a)=f(b)⇒a=b
Increasing and continuous are neither necessary nor the defining property; being one-to-one is.
Which function is its own inverse?
A self-inverse function satisfies f(f(x))=xf(f(x)) = xf(f(x))=x. Test 1x\frac{1}{x}x1.
f(f(x))=1 1/x =xf(f(x)) = \frac{1}{\,1/x\,} = xf(f(x))=1/x1=x
The others fail, and x2x^2x2 is not even one-to-one; only 1x\frac{1}{x}x1 is its own inverse.
To find a formula for f−1f^{-1}f−1, you can:
Write y=f(x)y = f(x)y=f(x), exchange xxx and yyy, and solve for the new yyy.
y=f(x) ⟶ x=f(y) ⟶ y=f−1(x)y = f(x) \;\longrightarrow\; x = f(y) \;\longrightarrow\; y = f^{-1}(x)y=f(x)⟶x=f(y)⟶y=f−1(x)
Taking a reciprocal, negating, or squaring does not invert a function.
Which function is the inverse of f(x)=x3f(x) = x^3f(x)=x3?
Undo cubing with the cube root. Solve y=x3y = x^3y=x3.
x=y3 ⇒ f−1(x)=x3x = \sqrt[3]{y} \;\Rightarrow\; f^{-1}(x) = \sqrt[3]{x}x=3y⇒f−1(x)=3x
The reciprocal 1x3\frac{1}{x^3}x31 is not the inverse.
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