Topic 2.2: Functions: Domain and Range Questions

The function g is defined for −2 ≤ x ≤ 5. The graph of g passes through a minimum at (1, −4) and a maximum at (5, 7).

f(x) = 3x − 5.

The mapping below shows a relationship between set A = {1, 2, 3} and set B = {4, 6, 8}. Each element of A maps to exactly one element of B.

A function is defined as f(x) = √(x − 3) for x ≥ 3.

f(x) = 4x − 7.

g(x) = (x + 2)/3.

f(x) = 2x + 1 for 0 ≤ x ≤ 4.

A phone-repair shop charges a fixed call-out fee plus an hourly rate. The total cost, in dollars, of a repair lasting t hours is modelled by C(t) = 35t + 20, for t ≥ 0.

(a) Find C(2.5).

(b) State what your answer to part (a) represents.

A function is defined by g(x) = 8/(x − 2) + 5, for 2 < x ≤ 6.

(a) Write down the value of x for which g is undefined.

(b) The inverse function g⁻¹ exists. Write down the range of g⁻¹.

The mass of a cooling sample, in grams, m minutes after measurement begins is modelled by M(m) = 40 × 0.85^m + 12, for m ≥ 0. Using your GDC, find M⁻¹(25), correct to three significant figures, and explain what it represents.

The temperature of a metal rod, in °C, m minutes after it is removed from a furnace is modelled by H(m) = 480 × 0.92^m + 20, for m ≥ 0.

(a) Write down the temperature of the rod at the instant it is removed from the furnace.

(b) Find H(15), correct to three significant figures.

The height of water in a draining tank, in metres, t minutes after a valve is opened is modelled by D(t) = 90/(t + 3), for 0 ≤ t ≤ 12.

Find the range of D.

A delivery service charges F(d) = 1.25d + 3.50 dollars to deliver a parcel a distance of d km, for d ≥ 0.

(a) Find F⁻¹(16).

(b) Interpret your answer to part (a) in context.