Functions: exam review Exam review: second paper One question at a time

The diagram shows the graphs of \(f(x)=e^x\) and \(g(x)=\ln x\), and the line \(y=x\). The point A, with x-coordinate 1.8, lies on the graph of f. [4]

1. (a) Write down the coordinates of A. Give the y-coordinate to 3 significant figures. [1]

   • A1 (1.8, 6.05)

2. (b) The point A is reflected in the line \(y=x\). Write down the coordinates of its image. [1]

   • A1 (6.05, 1.8)

3. (c) Write down the value of \(g(f(1.8))\). [1]

   • A1 1.8 (g undoes f)

4. (d) Write down how the functions f and g are related. [1]

   • R1 f and g are inverse functions (their graphs are reflections in y = x)

The function f is defined by \(f(x)=\dfrac{7x}{x-5}\), \(x\ne 5\). [5]

1. (a) Write down the equation of each asymptote of the graph of f. [2]

   • A1 x = 5
   • A1 y = 7

2. (b) The graph of \(g(x)=\dfrac{bx}{x-c}\) has asymptotes that meet at the point \((6, 2)\). Find b and c. [2]

   • A1 c = 6 (vertical asymptote x = c)
   • A1 b = 2 (horizontal asymptote y = b)

3. (c) Write down the range of f. [1]

   • A1 f(x) ≠ 7

The diagram shows the graphs of \(f(x)=10^x\) and \(g(x)=\log_{10}x\), and the line \(y=x\). The point A, with x-coordinate 0.9, lies on the graph of f. [4]

1. (a) Write down the coordinates of A. [1]

   • A1 (0.9, 7.94) (\(10^{0.9}=7.94\))

2. (b) Write down the coordinates of the point B on the graph of g that is the reflection of A, and the equation of the mirror line. [2]

   • A1 B(7.94, 0.9)
   • A1 y = x

3. (c) Write down the value of \(g(7.94)\). Give your answer to 3 significant figures. [1]

   • A1 0.900 (B is on the graph of g, so \(g(7.94)=0.9\))

The number of subscribers S of a video channel, t weeks after it starts, is modelled by \(S(t)=\dfrac{720t}{t+5}\), for \(t\ge0\). [4]

1. (a) Write down the equations of the vertical and horizontal asymptotes of the graph of \(S\). [2]

   • A1 t = −5
   • A1 S = 720

2. (b) Say what the horizontal asymptote tells you about the number of subscribers. [1]

   • A1 S tends to 720 as t grows, but never reaches it

3. (c) Find the number of subscribers after 10 weeks. [1]

   • A1 480 (\(720\times10\div15\))

A print shop makes poster prints. The cost \(C\) euros of an order of n posters is modelled by \(C(n)=3.2+0.45n\). [9]

1. (a) Find \(C(30)\), and find the value of n for which \(C(n)=21.2\). [3]

   • A1 C(30) = 16.70
   • M1 for 3.2 + 0.45n = 21.2
   • A1 n = 40

2. (b) The shop accepts at most 80 posters in one order. Write down a reasonable domain and a reasonable range for this model. [2]

   • A1 domain 0 ≤ n ≤ 80, n a whole number (you cannot order part of an item)
   • A1 range 3.2 ≤ C ≤ 39.20

3. (c) The function g is defined by \(g(x)=\sqrt{x+2}-1\). Write down the domain and the range of g. [2]

   • A1 x ≥ −2 (the number under the square root cannot be negative)
   • A1 g(x) ≥ −1 (a square root is at least 0)

4. (d) The function h is defined by \(h(x)=x^{2} - 4 x + 1\), for \(-1\le x\le 4\). Use your calculator to sketch the graph of h, and write down the range of h. [2]

   • M1 for the graph of h on the calculator with its end points and its lowest point
   • A1 −3 ≤ h(x) ≤ 6

A drone flies towards a landing pad that is 25 km from its base. Its distance D km from the base after t hours is modelled by \(D=a+bk^{-t}\), where \(k>1\). [10]

1. (a) Explain why a is equal to 25. [1]

   • R1 as t gets large, k^(−t) tends to 0, so D approaches a, the whole 25 km

2. (b) Find b. [2]

   • M1 for D(0) = 0, so a + b = 0
   • A1 b = −25

3. (c) Given that \(D(1)=8\), find the value of k. [3]

   • M1 for 25 − 25k^(−1) = 8
   • M1 for k^(−1) = 0.680 (17/25)
   • A1 k = 1.47

4. (d) Find the distance still to travel after 4 hours. [2]

   • M1 for 25 − D(4) or 25k^(−4)
   • A1 5.35

5. (e) Find the time at which the drone is 2 km from the end. Give your answer in hours to 3 significant figures. [2]

   • M1 for solving 25 − D(t) = 2, or 25k^(−t) = 2
   • A1 6.55

A cargo ferry sails towards an island that is 40 km from the harbour it leaves. Its distance D km from the start after t hours is modelled by \(D=a+bk^{-t}\), where \(k>1\). The ferry starts at \(D=0\), and after 1 hour it has travelled 15 km. [9]

1. (a) Explain why a is the total distance to the island, and write down the value of a. [2]

   • R1 as t increases, \(k^{-t}\) tends to 0, so D tends to a
   • A1 a = 40

2. (b) Find the value of b. [1]

   • A1 b = −40 (\(D(0)=0\): \(40+b=0\))

3. (c) Find the value of k. [2]

   • M1 for \(40-40k^{-1}=15\)
   • A1 k = 1.60

4. (d) Find how much further the ferry has to travel after 3 hours to reach the island. Give your answer to 3 significant figures. [2]

   • M1 for \(40\times1.60^{-3}\) (the distance remaining is \(40-D\))
   • A1 9.77

5. (e) Use your calculator to find the time, to the nearest hour, when the ferry has travelled 39.5 km. [2]

   • M1 for solving \(40-40\times1.60^{-t}=39.5\)
   • A1 9 (t = 9.32, to the nearest hour)

A pottery studio uses four models in its planning. [16]

1. (a) The profit \(P\) euros from selling mugs at x euros each is modelled by \(P(x)=- 2 x^{2} + 28 x - 48\). Use your calculator to find the y-intercept, the zeros and the vertex of the graph of P, and say what the vertex tells you about the profit. [4]

   • A1 y-intercept −48
   • A1 zeros x = 2 and x = 12
   • A1 vertex (7, 50)
   • R1 the greatest profit is 50 euros, made when the price is 7 euros

2. (b) The temperature of a mug taken from the kiln is modelled by \(T(t)=20+155e^{-0.04t}\), where T is in °C and t is in minutes after it leaves the kiln. Write down the equation of the horizontal asymptote of the graph of T, say what it means in this context, and find T after 20 minutes. Give your answer to 3 significant figures. [3]

   • A1 T = 20
   • R1 the temperature of the mug falls towards 20 °C (the temperature of the studio) but never goes below it
   • A1 89.6 °C

3. (c) The number N of mugs sold in a day is inversely proportional to the price p euros, so \(N=\dfrac{a}{p}\). When \(p=4\), \(N=60\). Find a, and find N when \(p=5\). Explain why the graph of N against p has the vertical axis as an asymptote. [4]

   • M1 for a = 60 × 4
   • A1 a = 240
   • A1 N = 48
   • R1 as p tends to 0, N becomes very large, so the graph gets close to the vertical axis but never meets it

4. (d) The cost per mug is modelled by \(C(n)=\dfrac{5n+90}{n+6}\), where n is the number made in a day. Find the y-intercept of the graph of C, and write down the equation of each asymptote. [3]

   • A1 y-intercept 15 (C(0) = 90/6)
   • A1 vertical asymptote n = −6
   • A1 horizontal asymptote C = 5

5. (e) The amount y of clay slip after h hours is measured: when \(h=0, 1, 2\), y is 3, 9, 27. Say whether a linear, a quadratic or an exponential model fits better, and justify your choice. [2]

   • A1 exponential
   • R1 y is multiplied by 3 each hour (a constant ratio), not increased by a constant amount