Domain and Range

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Domain and Range — Free MYP4 Mathematics (Extended) Practice Questions

1QuestionDomain from graphsConcept Practice

3 marks~5 minCriterion A

A solar panel technician monitors the power output of a panel. The output is modelled by the function

P = 3t - 1

, where

P

is the power output in watts and

t

is the time in hours after sunrise. The function is valid for

2 \leq t \leq 8

Calculate the minimum and maximum values of

P

for this domain. [2]

The panel must deliver at least 20 watts to charge a battery. Justify whether this condition is ever met within the given domain. [1]

Solutions

2QuestionDomain from graphsAssessment Practice

4 marks~6 minCriterion C

A city's water pressure monitoring system records pressure (in bar) as a piecewise linear function of time (in hours) over a 24-hour period. Due to scheduled maintenance, no data is recorded between

t = -1

and

t = 1

.

The graph shows two linear segments:
- Segment 1: from

(-4,\ 2)

(closed circle) to

(-1,\ 5)

(open circle)
- Segment 2: from

(1,\ 3)

(closed circle) to

(4,\ 0)

(closed circle)

Justify whether this graph represents a function. [1]

Deduce the domain of the function, expressing your answer in interval notation. [1]

The monitoring system requires at least 7 hours of combined recorded data to be considered reliable. Advise the network operator whether this session meets the reliability requirement, supporting your answer with calculations. [2]

Solutions

3QuestionRange from graphsAssessment Practice

4 marks~6 minCriterion B

A structural engineer models the vertical clearance beneath four cable-stay bridge sections using quadratic functions. Each function gives height

y

(metres) at horizontal position

x

(metres), with the vertex representing the lowest or highest point of the cable profile.

The four cable profiles are:

f(x) = x^2

, vertex

(0,\ 0)

, opens upward

g(x) = x^2 + 3

, vertex

(0,\ 3)

, opens upward

h(x) = (x-2)^2

, vertex

(2,\ 0)

, opens upward

k(x) = -(x+1)^2 + 4

, vertex

(-1,\ 4)

, opens downward

State the range of each function. [1]

Analyse the relationship between the vertex

y

-coordinate, the direction of opening, and the range of each profile. Deduce a general rule connecting the sign of

a

and the vertex

y

-coordinate

k

to the range of

a(x-h)^2 + k

. [2]

A new bridge section requires a cable profile

m(x) = a(x-h)^2 + k

whose height never exceeds

-3

m relative to the reference datum, so the range must be

(-\infty,\ -3]

. Construct a valid function

m(x)

satisfying this condition, and justify whether a cable profile with range

(-\infty,\ -3]

is physically safe for vehicles passing beneath the bridge. [1]

Solutions

4QuestionDomain from graphsAssessment Practice

6 marks~9 minCriterion D

The height

h

(in metres) of a projectile fired from the ground is modelled by

h(t) = -5t^2 + 40t

, where

t

is time in seconds. The graph shows a downward-opening parabola with vertex

(4,\ 80)

, crossing the

t

-axis at

t = 0

and

t = 8

. The horizontal axis shows

t

from 0 to 9 seconds; the vertical axis shows

h

from 0 to 90 metres.

State the domain of

h(t)

in this context. [1]

Using the graph, estimate the two values of

t

at which the projectile reaches a height of 75 m. [2]

A sensor records the projectile's height as 46.25 m at

t = 1.5

s. Calculate the percentage error between the model's prediction and the sensor reading, using

\text{percentage error} = \frac{|\text{predicted} - \text{actual}|}{|\text{actual}|} \times 100\%.

[2]

The safety system requires the model to be accurate to within 5\% at all times. Justify whether the model satisfies this requirement at

t = 1.5

s. [1]

Solutions

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