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Math Coach Amy
Amy Ferguson Moncure

Algebra 1 · Unit 6

Input and Output

Find f(x) for a given x, and find x when you are given f(x), from an equation, graph, or table.

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Key Ideas Easy Medium Hard Super Challenge
What to Remember

f(x) is the output

To find f(3), substitute 3 for x. The input is 3, the output is f(3).

Solve f(x) = k

Set the expression equal to k and solve for x. That x is in the domain if the story allows it.

From a graph or table

Read up from x to the graph to get f(x). Read across from a y-value to find x.

Check the story

An x that solves the algebra might still be outside the domain of the story.

Level 1

Easy

Evaluate f(x), or solve f(x) = k, for y = mx + b.

Example
If f(x) = 2x + 5, find f(3).

f(3) = 2(3) + 5 = 11

Answer: 11

1
If f(x) = −3x + 1, find f(2).
Show Answer

f(2) = −6 + 1 = −5

Answer: −5

2
If f(x) = 4x − 1, find x so that f(x) = 11.
Show Answer

4x − 1 = 11

4x = 12

x = 3

Answer: x = 3

3
The table shows f.
x1234
f(x)57911
Show Answer

f(3) = 9

Answer: 9

4
If g(x) = x − 8, find g(0) and the x with g(x) = 0.
Show Answer

g(0) = −8

x − 8 = 0, x = 8

Answer: g(0) = −8; x = 8

5
f(x) = 5. Find f(10) and f(−2).
Show Answer

A constant function. Both outputs are 5.

Answer: 5 and 5

Level 2

Medium

A graph, standard form, or a two-step solve.

Example
f(x) = 12x + 3. Find f(8) and the x with f(x) = 6.

f(8) = 4 + 3 = 7

12x + 3 = 6

12x = 3

x = 6

Answer: f(8) = 7; x = 6

1
The graph of y = −1x + 5 is shown. Find f(2) and the x with f(x) = 1.
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f(2) = 3

1 = −1x + 5, x = 4

Answer: 3; x = 4

2
2x + 3y = 12 defines y = f(x). Find f(3).
Show Answer

2(3) + 3y = 12

3y = 6

y = 2

f(3) = 2

Answer: 2

3
C(n) = 9n + 4 is the cost of n tickets. Find C(5) and the n with C(n) = 49.
Show Answer

C(5) = 45 + 4 = 49

9n + 4 = 49

n = 5

Five tickets cost 49 dollars.

Answer: 49; n = 5

4
f(x) = 3x − 7. Find f(f(3)).
Show Answer

f(3) = 2

f(2) = 6 − 7 = −1

Answer: −1

5
A table of a linear f:
x0246
f(x)15913
Show Answer

When f(x) = 9, x = 4.

Check: each +2 in x adds 4 to f, so m = 2, f(x) = 2x + 1.

Answer: x = 4

Level 3

Hard

A restricted domain, or an input that the story rejects.

Example
V(t) = 40 − 5t gallons, 0 ≤ t ≤ 8. Find V(3). Is there a t with V(t) = −5 in the story?

V(3) = 40 − 15 = 25

40 − 5t = −5

t = 9

t = 9 is outside 0 ≤ t ≤ 8. The tank is already empty at t = 8. The story has no such t.

Answer: 25; no

1
f(x) = 2x − 13. Find f(5) and x with f(x) = 3.
Show Answer

f(5) = 93 = 3

2x − 13 = 3

2x − 1 = 9

x = 5

Answer: 3; x = 5

2
A line through (0, 2) and (4, 10). Find f(6) without writing the equation first, then check with the equation.
Show Answer

m = 84 = 2

From x = 4 to x = 6 is +2, so y increases by 4: f(6) = 14

f(x) = 2x + 2

f(6) = 12 + 2 = 14

Answer: 14

3
f(x) = 4 − x on x = 0, 1, 2, 3, 4. Find every x with f(x) ≥ 2.
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4 − x ≥ 2

x ≤ 2

x = 0, 1, 2

Answer: 0, 1, 2

4
g(x) = 2x + 1. Solve g(x) = g(5).
Show Answer

2x + 1 = 11

x = 5

A linear function that is not constant is one-to-one, so only x = 5.

Answer: x = 5

5
A calculator table for f(x) = 1.5x + 2 lists f(4) = 8. Check by hand, and find x with f(x) = 11.
Show Answer

1.5(4) + 2 = 6 + 2 = 8. The table is right.

1.5x + 2 = 11

1.5x = 9

x = 6

Answer: x = 6

Super Challenge

Super Challenge

1
f(x) = mx + 3 and f(2) = 11. Find f(5).
Show Answer

2m + 3 = 11

m = 4

f(5) = 20 + 3 = 23

Answer: 23

2
On a graph, f(a) = b and f(b) = a with a ≠ b. If f is linear, what is the slope of the segment between those points, and what is f(f(a))?
Show Answer

Slope a − bb − a = −1

f(f(a)) = f(b) = a

Answer: slope −1; f(f(a)) = a

3
A parking fee is F(h) = 3h + 5 for h hours, h ≥ 0. Find F(2). Can F(h) = 2 in the story?
Show Answer

F(2) = 11

3h + 5 = 2

3h = −3

h = −1

Negative hours are not in the domain. No.

Answer: 11; no

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