12 multiple-choice questions, progressively harder.
The point (3,8)(3, 8)(3,8) lies on the graph of fff. Which point must lie on the graph of f−1f^{-1}f−1?
Solution
Correct answer: D
The graph of f−1f^{-1}f−1 is the reflection of the graph of fff across y=xy = xy=x, which swaps the two coordinates of every point.
(3,8)→(8,3)(3, 8) \to (8, 3)(3,8)→(8,3)
The input and output trade places, so (8,3)(8, 3)(8,3) is on f−1f^{-1}f−1.
If f(4)=9f(4) = 9f(4)=9, what is f−1(9)f^{-1}(9)f−1(9)?
The statement f(4)=9f(4) = 9f(4)=9 says the point (4,9)(4, 9)(4,9) is on the graph of fff, so its reflection (9,4)(9, 4)(9,4) is on the graph of f−1f^{-1}f−1.
f−1(9)=4f^{-1}(9) = 4f−1(9)=4
The inverse sends the output 999 back to the input 444.
The point (a,b)(a, b)(a,b) is on the graph of fff. Its reflection across y=xy = xy=x is
Correct answer: C
Reflection across y=xy = xy=x swaps both coordinates.
(a,b)→(b,a)(a, b) \to (b, a)(a,b)→(b,a)
Negating a single coordinate would be a reflection across an axis, not across y=xy = xy=x.
The point (−2,5)(-2, 5)(−2,5) lies on the graph of fff. Which point lies on f−1f^{-1}f−1?
Correct answer: B
Reflect across y=xy = xy=x by swapping the coordinates, keeping each sign attached to its number.
(−2,5)→(5,−2)(-2, 5) \to (5, -2)(−2,5)→(5,−2)
So (5,−2)(5, -2)(5,−2) is on the graph of f−1f^{-1}f−1.
A line fff has slope 222. Its inverse f−1f^{-1}f−1 is also a line. What is the slope of f−1f^{-1}f−1?
Correct answer: A
Reflecting across y=xy = xy=x swaps the horizontal run and vertical rise of the line, which turns the slope into its reciprocal.
slope of f−1=12\text{slope of } f^{-1} = \frac{1}{2}slope of f−1=21
A steep line of slope 222 reflects to a gentle line of slope 12\tfrac{1}{2}21.
A table gives f(1)=4f(1) = 4f(1)=4, f(2)=7f(2) = 7f(2)=7, and f(3)=10f(3) = 10f(3)=10. What is f−1(7)f^{-1}(7)f−1(7)?
The row f(2)=7f(2) = 7f(2)=7 gives the point (2,7)(2, 7)(2,7) on fff, so its swap (7,2)(7, 2)(7,2) is on f−1f^{-1}f−1.
f−1(7)=2f^{-1}(7) = 2f−1(7)=2
The inverse sends the output 777 back to the input 222.
Reflecting a graph across the line y=xy = xy=x swaps the roles of
The horizontal axis carries inputs and the vertical axis carries outputs; the reflection trades the two axes.
(input,output)→(output,input)(\text{input}, \text{output}) \to (\text{output}, \text{input})(input,output)→(output,input)
That is exactly what taking an inverse does.
A function fff has domain 0≤x≤80 \le x \le 80≤x≤8. What is the range of f−1f^{-1}f−1?
Reflecting across y=xy = xy=x trades the axes, so the range of the inverse is the domain of fff.
range of f−1=domain of f=0≤y≤8\text{range of } f^{-1} = \text{domain of } f = 0 \le y \le 8range of f−1=domain of f=0≤y≤8
The old inputs become the new outputs.
Which point lies on the line y=xy = xy=x?
A point is on y=xy = xy=x when its two coordinates are equal.
(4,4): y=x holds since 4=4(4, 4): \; y = x \text{ holds since } 4 = 4(4,4):y=x holds since 4=4
Such a point is its own reflection across the line.
The graph of fff passes through the origin (0,0)(0, 0)(0,0). Reflecting across y=xy = xy=x, the graph of f−1f^{-1}f−1 passes through
The origin lies on the line y=xy = xy=x, so it is a fixed point of the reflection.
(0,0)→(0,0)(0, 0) \to (0, 0)(0,0)→(0,0)
The reflected graph passes through the origin as well.
The point (9,2)(9, 2)(9,2) is on the graph of f−1f^{-1}f−1. What is f(2)f(2)f(2)?
The point (9,2)(9, 2)(9,2) on f−1f^{-1}f−1 reflects to (2,9)(2, 9)(2,9) on fff.
f(2)=9f(2) = 9f(2)=9
The inverse sends 999 back to 222, so fff sends 222 to 999.
To find a point on f−1f^{-1}f−1 from a point (a,b)(a, b)(a,b) on fff, you
The graph of f−1f^{-1}f−1 is the reflection of fff across y=xy = xy=x, and that reflection swaps coordinates.
Negating or doubling would move the point somewhere else entirely.
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