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Math Coach Amy
Amy Ferguson Moncure

Algebra 1 · Unit 2

Input and Output

From a linear equation or graph, find y for a given x, and find x for a given y.

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Key Ideas Easy Medium Hard Super Challenge
What to Remember

Input to output

Given x, substitute into the equation (or read up from x on the graph) to find y.

Output to input

Given y, set the expression equal to that number and solve for x (or read across from y on the graph).

Function notation

f(3) means the output when the input is 3. Solving f(x) = 11 means finding the input that gives output 11.

Check

The ordered pair (x, y) must lie on the line. Substitute to confirm.

Level 1

Easy

Substitute, or solve a one-step equation.

Example
If y = 3x + 1, find y when x = 4.

y = 3x + 1

Substitute x = 4.

y = 3(4) + 1

y = 12 + 1

y = 13

Answer: 13

1
If f(x) = 2x − 5, find f(6).
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f(6) = 2(6) − 5

f(6) = 12 − 5

f(6) = 7

Answer: 7

2
If y = −4x + 3, find y when x = 2.
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y = −4(2) + 3

y = −8 + 3

y = −5

Answer: −5

3
If y = 5x − 2, find x when y = 13.
Show Answer

13 = 5x − 2

15 = 5x

x = 3

Check: 5(3) − 2 = 13

Answer: x = 3

4
The graph of y = x + 2 is shown. Find y when x = 3, and find x when y = 0.
Show Answer

When x = 3, y = 5. Point (3, 5).

When y = 0, x = −2. Point (−2, 0).

Answer: y = 5; x = −2

5
Use the table to find y when x = 2, and x when y = 11.
x0123
y581114
Show Answer

When x = 2, y = 11.

When y = 11, x = 2.

That is the same pair (2, 11).

Answer: y = 11; x = 2

Level 2

Medium

Two-step solve, and a story.

Example
If f(x) = 12x − 4, find f(10) and find x so that f(x) = 1.

f(10) = 12(10) − 4 = 5 − 4 = 1

12x − 4 = 1

12x = 5

x = 10

The input 10 gives output 1, so both questions use (10, 1).

Answer: f(10) = 1; x = 10

1
A taxi is C = 0.80m + 3.50. Find the cost of a 6-mile ride, then find miles if the cost is $7.50.
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C(6) = 0.80(6) + 3.50 = 4.80 + 3.50 = 8.30

7.50 = 0.80m + 3.50

4 = 0.80m

m = 5

6 miles cost $8.30. $7.50 is a 5-mile ride.

Answer: $8.30; 5 miles

2
If g(x) = 3x + 7, solve g(x) = 1.
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3x + 7 = 1

3x = −6

x = −2

Check: 3(−2) + 7 = 1

Answer: x = −2

3
From the graph of y = −3x + 6, find the output at x = 1 and the input at y = 0.
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y = −3(1) + 6 = 3

0 = −3x + 6, x = 2

Answer: y = 3; x = 2

4
The table is linear. Find y when x = 5.
x1234
y0369
Show Answer

Each x-step of 1 increases y by 3, so m = 3.

When x = 4, y = 9, so x = 5 gives y = 12.

Or y = 3x − 3, and 3(5) − 3 = 12.

Answer: 12

5
If p(t) = 40 − 5t, find p(3) and find t when p(t) = 10. Interpret if t is minutes and p is gallons.
Show Answer

p(3) = 40 − 15 = 25

10 = 40 − 5t

5t = 30

t = 6

After 3 minutes, 25 gallons remain. 10 gallons remain after 6 minutes.

Answer: 25 gallons; t = 6

Level 3

Hard

Fractions, two functions, and reading a graph both ways.

Example
f(x) = 2x + 1 and g(x) = 13 − x. Find f(4), then find x so that f(x) = g(x).

f(4) = 8 + 1 = 9

2x + 1 = 13 − x

3x + 1 = 13

3x = 12

x = 4

So f(4) = g(4) = 9. The graphs meet at (4, 9).

Answer: f(4) = 9; they meet at x = 4

1
If h(x) = 23x − 5, find h(9) and find x so that h(x) = −1.
Show Answer

h(9) = 23(9) − 5 = 6 − 5 = 1

23x − 5 = −1

23x = 4

x = 6

Check: 23(6) − 5 = 4 − 5 = −1

Answer: h(9) = 1; x = 6

2
A line is shown through (0, 4) and (6, 1). Find the output at x = 2 and the input at y = 2.
Show Answer

m = 1 − 46 − 0 = −36 = −12

y = −12x + 4

At x = 2: y = −1 + 4 = 3

2 = −12x + 4

−21 = −12x

x = 4

Answer: y = 3; x = 4

3
Shop A: C = 0.10n + 22. Shop B: C = 0.25n + 10. For how many prints n is Shop A’s output equal to Shop B’s output?
Show Answer

0.10n + 22 = 0.25n + 10

12 = 0.15n

n = 80

Check: 0.10(80) + 22 = 30, and 0.25(80) + 10 = 30

Answer: n = 80

4
If k(x) = 5 − 2x, is there an input whose output is 11? Find it or explain.
Show Answer

11 = 5 − 2x

6 = −2x

x = −3

Yes. k(−3) = 5 + 6 = 11

Answer: x = −3

5
A graph of a linear function passes through (0, −2) with slope 3. Find the output at x = −1 and the input at y = 7.
Show Answer

y = 3x − 2

At x = −1: y = −3 − 2 = −5

7 = 3x − 2

9 = 3x

x = 3

Answer: y = −5; x = 3

Super Challenge

Super Challenge

1
A machine’s table only prints x = 0, 2, 4, 6 for y = 1.5x − 4. How can you find the output at x = 3 without a new table? How can you find the input for y = 5?
Show Answer

The rule is y = 1.5x − 4, so y(3) = 4.5 − 4 = 0.5.

Halfway between x = 2 (y = −1) and x = 4 (y = 2) is also 0.5, because the graph is a line.

5 = 1.5x − 4

9 = 1.5x

x = 6

That input is already a table row: when x = 6, y = 5.

Answer: y(3) = 0.5; x = 6

2
For y = mx + b with m ≠ 0, explain why every real output comes from exactly one input, and why that fails when m = 0 and you ask for an output other than b.
Show Answer

If m ≠ 0 you can solve x = y − bm. One y gives one x.

If m = 0, y is always b. The output b comes from every x. Any other output is impossible.

Horizontal lines fail the “one input for each output” idea (they are not one-to-one).

Answer: non-zero slope: one x per y; horizontal: only y = b

3
f(x) = 4x − 1. Student A computes f(2.5) on a calculator. Student B solves 4x − 1 = 9. Show they are asking different questions, then answer both.
Show Answer

f(2.5) = 4(2.5) − 1 = 10 − 1 = 9. That is input → output.

4x − 1 = 9 is output 9 → input. 4x = 10, x = 2.5.

Here the two questions happen to reverse each other because 9 is f(2.5). In general, “evaluate at 2.5” and “solve f(x) = 9” are opposite directions.

Answer: f(2.5) = 9; x = 2.5 when f(x) = 9

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