Task Sets: Read inverse from graph or table - Common Core Math - HS Functions

A

Recall / Warm-Up

1.

If $f(3) = 7$ , which of the following is true about $f^{-1}$ ?

A.

$f^{-1}(3) = 7$

B.

$f^{-1}(7) = 3$

C.

$f^{-1}(7) = \frac{1}{3}$

D.

$f^{-1}(3) = \frac{1}{7}$

2.

A table shows $f(1)=4$ , $f(2)=7$ , $f(3)=10$ . What is $f^{-1}(7)$ ?

A.

7

B.

2

C.

10

D.

3

3.

A function has these outputs: $f(1)=3$ , $f(2)=5$ , $f(3)=3$ . Does an inverse function exist?

A.

Yes — every input has exactly one output.

B.

No — the output 3 appears twice, so the function is not one-to-one.

C.

Yes — we can still swap the columns.

D.

No — none of the outputs are the same as the inputs.

B

Fluency Practice

1.

Use the table below to find $f^{-1}(14)$ .

$x$	$f(x)$
2	5
4	9
6	14
8	19

2.

Use the table below to find $f^{-1}(9)$ .

$x$	$f(x)$
1	3
3	9
5	15
7	21

3.

The graph of $f$ passes through the points $(1, 2)$ , $(3, 5)$ , $(5, 7)$ , and $(7, 9)$ . What is $f^{-1}(5)$ ?

4.

The graph of $f$ passes through $(-2, 1)$ , $(0, 3)$ , $(2, 5)$ , $(4, 7)$ . What is $f^{-1}(3)$ ?

5.

A table shows $g(1)=4$ , $g(2)=7$ , $g(3)=4$ , $g(4)=10$ . Does $g^{-1}$ exist as a function?

A.

Yes, because every input has exactly one output.

B.

No, because the output 4 appears for both inputs 1 and 3.

C.

Yes, because we can still list the pairs in reverse order.

D.

No, because the outputs don't include every integer.

C

Varied Practice

$x$	$h(x)$
0	6
2	10
4	14
6	18

D

Word Problems

1.

A factory's production table shows items produced after $t$ hours of work:

Hours ( $t$ )	Items produced $f(t)$
1	15
2	30
3	45
4	60

1.

How many hours does it take to produce 45 items?

2.

Write one sentence explaining what $f^{-1}(60)$ means in this context and give its value.

2.

A thermometer records temperature $T$ (in °F) at time $t$ (hours after midnight): $T(0)=58$ , $T(2)=62$ , $T(4)=70$ , $T(6)=75$ , $T(8)=80$ .

At what time (hours after midnight) did the temperature reach 70°F? That is, find $T^{-1}(70)$ .

E

Error Analysis

1.

A table shows $f(1)=3$ , $f(2)=7$ , $f(3)=11$ , $f(4)=15$ .

Student work: "To find $f^{-1}(2)$ , I look at row $x=2$ and read $f(2) = 7$ . So $f^{-1}(2) = 7$ ."

What error did the student make?

A.

The student evaluated $f(2)$ instead of $f^{-1}(2)$ . They should search the output column for 2, but 2 is not an output, so $f^{-1}(2)$ cannot be determined from this table.

B.

The student forgot to check whether the function is one-to-one before finding the inverse.

C.

The student should have computed $\frac{1}{f(2)} = \frac{1}{7}$ , not 7.

D.

The student used the wrong row; they should use $x = 7$ instead.

2.

A table shows $p(0)=5$ , $p(1)=8$ , $p(2)=5$ , $p(3)=12$ .

Student work: "I swap the columns to build the inverse table. The inverse maps 5 to 0, 8 to 1, 5 to 2, 12 to 3. So $p^{-1}$ maps 5 to 0 and also 5 to 2."

What should the student conclude, and why?

F

Challenge / Extension

1.

The graph of $f$ passes through the following points: $(1, 3)$ , $(2, 5)$ , $(3, 7)$ , $(4, 9)$ .

If the graph of $f^{-1}$ is sketched by reflecting $f$ over the line $y = x$ , what is the $y$ -coordinate of the point on $f^{-1}$ with $x$ -coordinate 7?

2.