Read Inverses from Graphs or Tables - Algebra 2

1. From the graph shown, if $f^{-1}(1) = -2$, what is the value of $f(-2) \cdot f^{-1}(-1)$?
   Answer: $3$ Given $f^{-1}(1) = -2$, we know $f(-2) = 1$. From the graph, we can read that $f(3) = -1$, so $f^{-1}(-1) = 3$. Therefore, $f(-2) \cdot f^{-1}(-1) = 1 \cdot 3 = 3$. Choice A results from using $f^{-1}(-1) = -3$. Choice B results from computing $f^{-1}(1) + f(0) = -2 + 4$. Choice D results from using $f(-2) = -2$ instead of 1.
2. The table gives values of a one-to-one function $f$, so $f$ has an inverse. Use the table to find $f^{-1}(-4)$. (If $(a,b)$ is on $f$, then $(b,a)$ is on $f^{-1}$.)

   $x$	$f(x)$
   -3	5
   -1	1
   0	-2
   2	-4
   4	-7

   Answer: $f^{-1}(-4)=2$ This question tests your ability to read values of an inverse function from a table or graph using the fundamental property that inverses swap inputs and outputs. If f has a table or graph showing its (x, y) pairs, then f⁻¹ has the same pairs but swapped: every (x, y) on f becomes (y, x) on f⁻¹. So to find f⁻¹(a), you don't need the inverse's table or graph—just find where y = a in f's representation and read the corresponding x-value. That x is your answer! For example, if f's table shows x = 3 gives y = 7, then f⁻¹(7) = 3. The inverse reverses the input-output relationship. To find f⁻¹(-4), I need to locate -4 in the f(x) column. Scanning down: f(-3) = 5, f(-1) = 1, f(0) = -2, f(2) = -4, f(4) = -7. I found it! When x = 2, f(x) = -4. Therefore, f⁻¹(-4) = 2. Choice A correctly finds f⁻¹(-4) = 2 by identifying that f(2) = -4 in the table. Choice B incorrectly assumes f⁻¹(-4) = -4, missing the fundamental swap property of inverses. Table reading strategy for f⁻¹(a): (1) Look in the f(x) column (the outputs) for the value a, (2) When you find a, look at the x-value in that same row, (3) That x-value is f⁻¹(a). Remember: you're working backwards from output to input!
3. The function $f$ is one-to-one, so it has an inverse. Using the table, find $f^{-1}(2)$.

   $x$	$f(x)$
   $-2$	$-4$
   $0$	$-1$
   $1$	$2$
   $3$	$5$
   $6$	$9$

   Answer: $1$ This question tests your ability to read values of an inverse function from a table or graph using the fundamental property that inverses swap inputs and outputs. If f has a table or graph showing its (x, y) pairs, then f⁻¹ has the same pairs but swapped: every (x, y) on f becomes (y, x) on f⁻¹. So to find f⁻¹(a), you don't need the inverse's table or graph—just find where y = a in f's representation and read the corresponding x-value. That x is your answer! For example, if f's table shows x = 3 gives y = 7, then f⁻¹(7) = 3. The inverse reverses the input-output relationship. To find $f^{-1}$(2), look in the f(x) column for the value 2; you'll see it's paired with x=1 in the table. Choice C correctly finds f⁻¹(2) = 1 by reading from the table properly. Some students might mistakenly pick 2 itself or confuse with another row, but always focus on locating the output first and grabbing the input— you're doing great, keep going! Table reading strategy for f⁻¹(a): (1) Look in the f(x) column (the outputs) for the value a, (2) When you find a, look at the x-value in that same row, (3) That x-value is f⁻¹(a). Example: table shows x: 1, 2, 3, 4 and f(x): 5, 9, 13, 17. To find f⁻¹(13), scan the f(x) column for 13 (found in row 3), read x from that row (x = 3), so f⁻¹(13) = 3. The swap happens automatically when you read this way!
4. The function $f$ is one-to-one, so it has an inverse. A point on the graph of $f$ is $(7,-3)$. Using the fact that $(a,b)$ on $f$ corresponds to $(b,a)$ on $f^{-1}$, what is $f^{-1}(-3)$?
   Answer: $f^{-1}(-3)=7$ This question tests your ability to read values of an inverse function from given information using the fundamental property that inverses swap inputs and outputs. If f has a point (a, b), then f⁻¹ has the point (b, a)—every (x, y) on f becomes (y, x) on f⁻¹. So to find f⁻¹(b), you look for the point on f where y = b and read the x-coordinate. Given that (7, -3) is on the graph of f, this means f(7) = -3. By the inverse relationship, this tells us that f⁻¹(-3) = 7. The coordinates swap! Choice B correctly identifies f⁻¹(-3) = 7 by recognizing that the point (7, -3) on f becomes the point (-3, 7) on f⁻¹. Choice A incorrectly suggests f⁻¹(-3) = -3, which would require the point (-3, -3) to be on f. Remember: the inverse function reverses the input-output relationship. If f takes 7 to -3, then f⁻¹ takes -3 back to 7. This swapping property is fundamental to understanding inverse functions!
5. A one-to-one function $f$ is given by the table, so $f$ has an inverse. What is $f^{-1}(14)$?

   $x$	$f(x)$
   $1$	$-2$
   $2$	$4$
   $3$	$9$
   $4$	$14$
   $5$	$20$

   Answer: $f^{-1}(14)=4$ This question tests your ability to read values of an inverse function from a table or graph using the fundamental property that inverses swap inputs and outputs. If f has a table or graph showing its (x, y) pairs, then f⁻¹ has the same pairs but swapped: every (x, y) on f becomes (y, x) on f⁻¹. So to find f⁻¹(a), you don't need the inverse's table or graph—just find where y = a in f's representation and read the corresponding x-value. That x is your answer! For example, if f's table shows x = 3 gives y = 7, then f⁻¹(7) = 3. The inverse reverses the input-output relationship. Looking for f⁻¹(14), I scan the f(x) column and find f(4) = 14. This means when x = 4, the output is 14, so f⁻¹(14) = 4. Choice B correctly finds f⁻¹(14) = 4 by reading from the table that f(4) = 14 and applying the swap principle. Choice D incorrectly reads 20 from the table, which is actually f(5), not related to finding f⁻¹(14). Table reading strategy for f⁻¹(a): (1) Look in the f(x) column (the outputs) for the value a, (2) When you find a, look at the x-value in that same row, (3) That x-value is f⁻¹(a). Remember, you're working backwards—from output to input—which is exactly what an inverse function does!
6. The function $f$ is one-to-one and has an inverse. Use the table to find $f^{-1}(0)$.

   $x$	$f(x)$
   $-3$	$-5$
   $-1$	$-2$
   $2$	$0$
   $4$	$3$
   $7$	$8$

   Answer: $2$ This question tests your ability to read values of an inverse function from a table or graph using the fundamental property that inverses swap inputs and outputs. If f has a table or graph showing its (x, y) pairs, then f⁻¹ has the same pairs but swapped: every (x, y) on f becomes (y, x) on f⁻¹. So to find f⁻¹(a), you don't need the inverse's table or graph—just find where y = a in f's representation and read the corresponding x-value. That x is your answer! For example, if f's table shows x = 3 gives y = 7, then f⁻¹(7) = 3. The inverse reverses the input-output relationship. To find $f^{-1}$(0), look in the f(x) column for the value 0; you'll see it's paired with x=2 in the table. Choice B correctly finds f⁻¹(0) = 2 by reading from the table properly. One might accidentally pick 0 or a negative, but recall we're reversing: find input for that output— you're making fantastic progress! Table reading strategy for f⁻¹(a): (1) Look in the f(x) column (the outputs) for the value a, (2) When you find a, look at the x-value in that same row, (3) That x-value is f⁻¹(a). Example: table shows x: 1, 2, 3, 4 and f(x): 5, 9, 13, 17. To find f⁻¹(13), scan the f(x) column for 13 (found in row 3), read x from that row (x = 3), so f⁻¹(13) = 3. The swap happens automatically when you read this way!
7. Function $k$ has an inverse, and the following values are known: $k(1) = 3$, $k(4) = -2$, and $k(7) = 0$. What is $k^{-1}(-2) + k^{-1}(0)$?
   Answer: $11$ From $k(4) = -2$, we get $k^{-1}(-2) = 4$. From $k(7) = 0$, we get $k^{-1}(0) = 7$. Therefore, $k^{-1}(-2) + k^{-1}(0) = 4 + 7 = 11$. Choice B results from adding the output values $-2 + 0$. Choice C results from using $k(1) = 3$ incorrectly. Choice D results from computing $k^{-1}(-2) + k^{-1}(3) = 4 + 1$.
8. The table gives outputs of a function $f$ that has an inverse. Since points swap, $(x, f(x))$ on $f$ corresponds to $(f(x), x)$ on $f^{-1}$. Use the table to find $f^{-1}(13)$.

   $x$	$f(x)$
   2	1
   4	6
   6	10
   7	13
   9	18

   Answer: $7$ This question tests your ability to read values of an inverse function from a table or graph using the fundamental property that inverses swap inputs and outputs. If f has a table or graph showing its (x, y) pairs, then f⁻¹ has the same pairs but swapped: every (x, y) on f becomes (y, x) on f⁻¹. So to find f⁻¹(a), you don't need the inverse's table or graph—just find where y = a in f's representation and read the corresponding x-value. That x is your answer! For example, if f's table shows x = 3 gives y = 7, then f⁻¹(7) = 3. The inverse reverses the input-output relationship. To find $f^{-1}$(13), look in the f(x) column for the value 13; you'll see it's paired with x=7 in the table. Choice C correctly finds f⁻¹(13) = 7 by reading from the table properly. It's easy to misread and pick a nearby value like 6 or 9, but always verify the exact match in the output column— excellent work persisting through this! Table reading strategy for f⁻¹(a): (1) Look in the f(x) column (the outputs) for the value a, (2) When you find a, look at the x-value in that same row, (3) That x-value is f⁻¹(a). Example: table shows x: 1, 2, 3, 4 and f(x): 5, 9, 13, 17. To find f⁻¹(13), scan the f(x) column for 13 (found in row 3), read x from that row (x = 3), so f⁻¹(13) = 3. The swap happens automatically when you read this way!
9. A one-to-one function $f$ is shown by the table, so $f$ has an inverse. Use the table to find $f^{-1}(12)$. (Look for the output 12 in the $f(x)$ column; the corresponding $x$ is the inverse value.)

   $x$	$f(x)$
   0	1
   3	6
   5	10
   6	12
   8	16

   Answer: $f^{-1}(12)=6$ This question tests your ability to read values of an inverse function from a table or graph using the fundamental property that inverses swap inputs and outputs. If f has a table or graph showing its (x, y) pairs, then f⁻¹ has the same pairs but swapped: every (x, y) on f becomes (y, x) on f⁻¹. So to find f⁻¹(a), you don't need the inverse's table or graph—just find where y = a in f's representation and read the corresponding x-value. That x is your answer! For example, if f's table shows x = 3 gives y = 7, then f⁻¹(7) = 3. The inverse reverses the input-output relationship. Looking for f⁻¹(12), I scan the f(x) column for the value 12: f(0) = 1, f(3) = 6, f(5) = 10, f(6) = 12, f(8) = 16. Found it! When x = 6, f(x) = 12. Therefore, f⁻¹(12) = 6. Choice C correctly finds f⁻¹(12) = 6 by locating the row where f(x) = 12 and reading the corresponding x-value. Choice A incorrectly assumes f⁻¹(12) = 12, which would require f(12) = 12, but 12 isn't even an input in the table. Table reading strategy for f⁻¹(a): (1) Look in the f(x) column (the outputs) for the value a, (2) When you find a, look at the x-value in that same row, (3) That x-value is f⁻¹(a). The process naturally reverses the function's action!
10. A one-to-one function $f$ is shown by the table, so $f$ has an inverse. Use the table to evaluate $f^{-1}(21)$. (Because inverse pairs swap, if $f(x)=21$ then $f^{-1}(21)=x$.)

    $x$	$f(x)$
    2	9
    4	13
    6	17
    8	21
    10	25

    Answer: $f^{-1}(21)=8$ This question tests your ability to read values of an inverse function from a table or graph using the fundamental property that inverses swap inputs and outputs. If f has a table or graph showing its (x, y) pairs, then f⁻¹ has the same pairs but swapped: every (x, y) on f becomes (y, x) on f⁻¹. So to find f⁻¹(a), you don't need the inverse's table or graph—just find where y = a in f's representation and read the corresponding x-value. That x is your answer! For example, if f's table shows x = 3 gives y = 7, then f⁻¹(7) = 3. The inverse reverses the input-output relationship. To evaluate f⁻¹(21), I scan the f(x) column for 21: f(2) = 9, f(4) = 13, f(6) = 17, f(8) = 21, f(10) = 25. Found it! When x = 8, f(x) = 21. Therefore, f⁻¹(21) = 8. Choice C correctly finds f⁻¹(21) = 8 by locating where f(x) = 21 in the table. Choice A incorrectly assumes f⁻¹(21) = 21, which would require f(21) = 21, but 21 isn't even an input value in the table. Table reading strategy for f⁻¹(a): (1) Look in the f(x) column (the outputs) for the value a, (2) When you find a, look at the x-value in that same row, (3) That x-value is f⁻¹(a). The swap happens automatically when you read this way!
11. The table shows some values of function $p(x)$. If $p$ has an inverse, which of the following could be the value of $p(6)$?
    Answer: $9$ For $p$ to have an inverse, each output value must correspond to exactly one input value. The table shows outputs 1, 3, and 5 are already used, so $p(6)$ cannot equal any of these values. Only 9 is not already an output value in the table. Choices A, B, and C would create repeated output values, violating the requirement for an inverse function.
12. The graph shows function $h(x)$. If $h^{-1}(4) = 2$, which point must be on the graph of $h(x)$?
    Answer: $(2, 4)$ If $h^{-1}(4) = 2$, then by definition of inverse functions, $h(2) = 4$. This means the point $(2, 4)$ is on the graph of $h(x)$. Choice B confuses the roles of input and output. Choice C uses the wrong sign for the x-coordinate. Choice D uses the wrong sign for the y-coordinate.
13. Based on the table shown, what is the value of $g^{-1}(5)$?
    Answer: $3$ To find $g^{-1}(5)$, we need to find the input value that produces an output of 5. From the table, $g(3) = 5$, so $g^{-1}(5) = 3$. Choice A gives $g^{-1}(-1)$ instead of $g^{-1}(5)$. Choice B gives $g^{-1}(8)$. Choice D gives $g(5)$ instead of $g^{-1}(5)$.
14. Functions $s$ and $t$ are inverses of each other. If $s(3) = 7$ and $s(5) = 2$, what is the value of $t(7) - t(2)$?
    Answer: $-2$ Since $s$ and $t$ are inverses, $t = s^{-1}$. From $s(3) = 7$, we get $t(7) = s^{-1}(7) = 3$. From $s(5) = 2$, we get $t(2) = s^{-1}(2) = 5$. Therefore, $t(7) - t(2) = 3 - 5 = -2$. Choice B gives $|t(7) - t(2)|$. Choice C gives $t(2) - t(7)$. Choice D gives $t(7) + t(2) + 3$.
15. The graph shows function $r(x)$. Based on the graph, what is $r^{-1}(0) + r^{-1}(2)$?
    Answer: $4$ To find $r^{-1}(0)$, locate where the graph has a y-value of 0, which occurs at $x = -1$. So $r^{-1}(0) = -1$. To find $r^{-1}(2)$, locate where the graph has a y-value of 2, which occurs at $x = 5$. So $r^{-1}(2) = 5$. Therefore, $r^{-1}(0) + r^{-1}(2) = -1 + 5 = 4$. Choice A results from adding the output values $0 + 2 - 1$. Choice B results from counting intercepts incorrectly. Choice D results from using $r^{-1}(2) = 4$ instead of 5.

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