Function Models | Math Coach Amy

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Key Ideas Easy Medium Hard

Substitute, then round only when the result is not exact

What to Remember

Read the input

t = 0 is the start. Substitute before you reach for a calculator.

Three significant figures

When the power is not a whole number that simplifies, round the decimal to 3 significant figures.

Domain and range still apply

A model can ask for the largest domain or the range of the rule, not only a single output.

A limit of the model

Say what real constraint the formula ignores, such as a top speed, a full pond, or a limited number of customers.

Easy

A linear model.

Example

A tank drains so the volume, in litres, is V(t) = 80 − 5t. Find the volume after 6 minutes.

V(t) = 80 − 5t

V(6) = 80 − 5(6)

V(6) = 80 − 30

V(6) = 50

Answer: 50 litres

The speed of a cart, in metres per second, is v(t) = 4t. Find v(3).

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v(x) = 4x

v(3) = 4(3)

v(3) = 12

Answer: 12 m/s

f(x) = 6x − 4. Find f(2), then solve f(x) = 20.

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f(x) = 6x − 4

f(2) = 6(2) − 4

f(2) = 8

Now solve 6x − 4 = 20.

6x = 24

x = 4

Answer: f(2) = 8, x = 4

A shuttle charges $9 plus $1 for every 4 km. Write C for k kilometres, then find the cost of 20 km.

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C = 9 + k4

C = 9 + 204

C = 9 + 5

C = 14

Answer: C = 9 + k4, $14

The temperature in a freezer, in °C, is T(h) = 8 − 2h. Find T(3).

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f(x) = −2x + 8

f(3) = −2(3) + 8

f(3) = 2

Answer: 2

A printer has already made 40 pages and then prints 12 pages each minute. Write P(m) and find P(5).

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P(m) = 40 + 12m

P(5) = 40 + 12(5)

P(5) = 100

Answer: P(m) = 40 + 12m, 100 pages

Medium

A root, a squared denominator, or an exponential.

Example

A pond starts with some fish. After t months, a model says N = 140 − 45e^−0.25t. How many fish does the model give at the start? How many does it give after 8 months, to 3 significant figures? Give one reason the model is not perfect.

N = 140 − 45e^−0.25t

At t = 0, the power is 0, and e⁰ = 1.

N = 140 − 45

N = 95

At t = 8, N = 140 − 45e⁻²

N is about 134

A real pond can lose fish to disease or fishing, and it cannot hold a population that the model keeps pushing toward 140 forever.

Answer: 95 at the start, about 134 after 8 months

f(x) = 6(x − 2)². Write the largest possible domain, find f(4), and state the range.

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The denominator is 0 when x − 2 = 0.

The domain is x ≠ 2.

f(4) = 64

f(4) = 32

A square is positive except at x = 2, where f is undefined. The fraction is always positive and can be any positive number.

The range is y > 0.

Answer: x ≠ 2, f(4) = 32, y > 0

f(x) = square root of 3x − 6. Find the largest domain, the range, and the solution of f(x) = 3.

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f(x) = square root of 3x − 6

The inside of a square root must be at least 0.

3x − 6 ≥ 0

3x ≥ 6

x ≥ 2

The range is y ≥ 0.

square root of 3x − 6 = 3

3x − 6 = 9

3x = 15

x = 5

Answer: x ≥ 2, y ≥ 0, x = 5

A company models tablet sales, in millions, by S = 1.6e^0.05t + 0.7, where t is years from now. Predict S after 4 years, to 3 significant figures, and give one reason a 40-year prediction may fail.

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S = 1.6e^{0.05 · 4} + 0.7

S = 1.6e^0.2 + 0.7

S is about 2.65

The model keeps growing forever. Materials, customers, and fashion will not do that for 40 years.

Answer: about 2.65 million; unlimited growth is not realistic

A shuttle charges $9 plus $1 for every 4 kilometres. Write C for a trip of k kilometres, then find the cost of 20 km.

Show Answer

C = 9 + k4

C = 9 + 204

C = 9 + 5

C = 14

Answer: C = 9 + k4, $14

A model says the speed of a drone, in metres per second, is v(t) = 3.5t. Find the speed after 4 seconds. Why is the model a poor prediction for t = 50?

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v(t) = 3.5t

v(4) = 3.5 × 4

v(4) = 14

At 50 seconds the model gives 175 m/s, faster than this drone can fly.

Answer: 14 m/s; the model has no maximum speed

Hard

Put a domain, a value, and a limitation together.

Example

f(x) = square root of 3x − 6. Find the largest domain, the range, and the solution of f(x) = 3.

f(x) = square root of 3x − 6

The inside of a square root must be at least 0.

3x − 6 ≥ 0

3x ≥ 6

x ≥ 2

The range is y ≥ 0.

square root of 3x − 6 = 3

3x − 6 = 9

3x = 15

x = 5

Answer: x ≥ 2, y ≥ 0, x = 5

Show Answer

N = 140 − 45e^−0.25t

At t = 0, the power is 0, and e⁰ = 1.

N = 140 − 45

N = 95

At t = 8, N = 140 − 45e⁻²

N is about 134

A real pond can lose fish to disease or fishing, and it cannot hold a population that the model keeps pushing toward 140 forever.

Answer: 95 at the start, about 134 after 8 months

A company models tablet sales, in millions, by S = 1.6e^0.05t + 0.7, where t is years from now. Predict S after 4 years, to 3 significant figures, and give one reason a 40-year prediction may fail.

Show Answer

S = 1.6e^{0.05 · 4} + 0.7

S = 1.6e^0.2 + 0.7

S is about 2.65

The model keeps growing forever. Materials, customers, and fashion will not do that for 40 years.

Answer: about 2.65 million; unlimited growth is not realistic

f(x) = 6(x − 2)². Write the largest possible domain, find f(4), and state the range.

Show Answer

The denominator is 0 when x − 2 = 0.

The domain is x ≠ 2.

f(4) = 64

f(4) = 32

A square is positive except at x = 2, where f is undefined. The fraction is always positive and can be any positive number.

The range is y > 0.

Answer: x ≠ 2, f(4) = 32, y > 0

One pound is worth 1.25 dollars. Let x be pounds, and let f(x) be dollars. Write f(x) and f⁻¹(x).

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f(x) = 54x

f⁻¹(x) = 45x

f⁻¹ turns dollars back into pounds.

Answer: f(x) = 54x, f⁻¹(x) = 45x

A model says the speed of a drone, in metres per second, is v(t) = 3.5t. Find the speed after 4 seconds. Why is the model a poor prediction for t = 50?

Show Answer

v(t) = 3.5t

v(4) = 3.5 × 4

v(4) = 14

At 50 seconds the model gives 175 m/s, faster than this drone can fly.

Answer: 14 m/s; the model has no maximum speed

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