Domain and Range

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Domain and Range — Free MYP5 Mathematics (Extended) Practice Questions

1QuestionRestrictionsConcept Practice

4 marks~6 minCriterion A

A chemical plant monitors the concentration of a coolant in a pipe. The concentration (in grams per litre) at time

x

hours after startup is modelled by

f(x) = \frac{4}{x - 3} + 2, \quad x > 0.

The function has a vertical asymptote at

x = 3

and a horizontal asymptote at

y = 2

Explain how each asymptote restricts the domain and range of

f

. [2]

Calculate

f(5)

. [1]

The pipe is safe to operate only when the concentration exceeds 3 g/L. Justify whether the pipe is safe to operate at

x = 5

hours. [1]

Solutions

2QuestionRange from graphsAssessment Practice

3 marks~5 minCriterion D

A drone delivery company models the height of a package above ground during a short flight using the function

h(t) = t^2 - 4t + 3

, where

h

is the height in metres and

t

is the time in seconds. The graph of

h(t)

shows a parabola with vertex

(2, -1)

that opens upwards.

Deduce the range of

h(t)

. [1]

The package must remain at or above ground level (

h \geq 0

) throughout the flight. Justify whether this model is physically valid, using your answer to part (a). [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

4QuestionRange from graphsAssessment Practice

5 marks~8 minCriterion C

A hospital monitoring system records a patient's oxygen saturation level,

f(x)

(%), over a 24-hour period. The piecewise function

f(x)

is represented by the graph below.

The graph consists of three parts:
- a horizontal segment at

y = 4

for

-5 \leq x < -2

, with a closed circle at

(-5, 4)

and an open circle at

(-2, 4)

- a downward-opening parabola with vertex

(0, 3)

, passing through

(-2, 1)

and

(2, 1)

, with closed circles at both endpoints
- a curve for

x > 2

starting at

(2, 1)

with a closed circle, decreasing toward the asymptote

y = -1

x \to \infty

, never reaching

y = -1

State the

y

-value produced by the horizontal segment and identify whether it is included in the range. [1]

Deduce the set of

y

-values contributed by the parabola and by the asymptotic curve, showing your reasoning for each endpoint. [2]

Interpret the complete range of

f(x)

in interval notation, and justify whether the gap in the range gives cause for clinical concern about the reliability of the monitoring data. [2]

Solutions

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