Functions: exam review Exam review: second paper Worked solutions

The functions f, g and h are defined by \(f(x)=2 x - 5\), \(g(x)=x^{2} + 3\) and \(h(x)=\sqrt{x - 1}\). [6]

1. (a) Find \(f(g(x))\). Give your answer in expanded form. [2]

   • M1 for substituting g(x) into f
   • A1 2x^2 + 1

2. (b) Find \(g(f(x))\). Give your answer in expanded form. [2]

   • M1 for substituting f(x) into g
   • A1 4x^2 − 20x + 28

3. (c) Find the largest possible domain of \(h(f(x))\). [2]

   • M1 for h(f(x)) = √(2x − 6), and 2x − 6 ≥ 0
   • A1 x ≥ 3

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 \(f(x)=\dfrac{7x-4}{2}\). [5]

1. (a) Find \(f^{-1}(x)\). [2]

   • M1 for swapping x and y and making y the subject: x = (7y − 4)/2
   • A1 f⁻¹(x) = (2x + 4)/7

2. (b) Show that \(f(f^{-1}(x))=x\). [3]

   • M1 for substituting f⁻¹(x) into f
   • A1 f(f⁻¹(x)) = (7×(2x + 4)/7 − 4)/2
   • A1 = 2x/2 = x

The function f is defined by \(f(x)=x^{3} - 5 x^{2} + 3\). The function is not one-to-one for all x. Use your calculator where it helps. [5]

1. (a) Find the least value of k such that f is one-to-one for \(x\ge k\). Give k to 3 significant figures. [2]

   • M1 for finding the minimum point of the graph, using the calculator or f′(x) = 0
   • A1 k = 3.33

2. (b) Write down the range of f for the domain \(x\ge k\). [1]

   • A1 f(x) ≥ −15.5

3. (c) Find \(f^{-1}(-13)\). [2]

   • M1 for solving f(x) = −13 for x ≥ k
   • A1 4

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

The functions f and g are defined by \(f(x)=3 x - 1\) and \(g(x)=x^{2} + 2\). [7]

1. (a) The function h is defined by \(h(x)=\sqrt{x^{2}+5}\). Write h in the form \(h(x)=p(q(x))\), and give the functions p and q. [2]

   • A1 q(x) = x² + 5 (the inside function)
   • A1 p(x) = √x (the outside function)

2. (b) Solve \(f(g(x))=32\). [2]

   • M1 for f(g(x)) = 3x² + 5
   • A1 x = −3 or x = 3

3. (c) Solve \(f(g(x))=g(f(x))\). Give your answers to 3 significant figures. [3]

   • M1 for g(f(x)) = 9x² - 6x + 3 and f(g(x)) = g(f(x))
   • M1 for solving the quadratic equation (calculator or formula)
   • A1 x = −0.264 or x = 1.26

Each part of this question is about a graph and its transformations. [9]

1. (a) The graph of \(y=x^2-5x\) is translated 2 units to the left. Find the equation of the new graph in the form \(y=ax^2+bx+c\). [2]

   • M1 for replacing x by (x + 2) in y = x² − 5x
   • A1 y = x² − x − 6

2. (b) The graph of \(y=2^x\) is stretched vertically with scale factor 3, and then translated 4 units up. Write down the equation of the new graph and the equation of its asymptote. [3]

   • M1 for stretching first: 3(2^x)
   • A1 y = 3(2^x) + 4
   • A1 asymptote y = 4 (the asymptote y = 0 is stretched to y = 0 and then moved 4 up)

3. (c) The point \((6, -4)\) lies on the graph of \(y=f(x)\). The graph of \(y=-2f(-3x)\) is drawn. Describe the two stretches and the two reflections that map the graph of \(y=f(x)\) onto it, and write down the image of the point. [4]

   • A1 vertical stretch, scale factor 2, with a reflection in the x-axis
   • A1 horizontal stretch, scale factor 1/3, with a reflection in the y-axis
   • M1 for x-coordinate 6 ÷ (−3) and y-coordinate (−2) × (−4)
   • A1 (−2, 8)

The functions f and g are defined by \(f(x)=x^{2} - 4 x\) and \(g(x)=x^{2} - 5 x\). Their graphs are shown. [12]

1. (a) Use the graphs to find \(f(g(6))\). [2]

   • M1 for reading g(6) = 6 from the graph of g
   • A1 f(6) = 12

2. (b) Use the graphs to find \(g(f(5))\). [2]

   • M1 for reading f(5) = 5 from the graph of f
   • A1 g(5) = 0

3. (c) Solve \(g(f(x))=0\). [4]

   • M1 for g(u) = 0 where u = f(x): u = 0 or 5
   • M1 for solving f(x) = 0: x = 0, 4
   • M1 for solving f(x) = 5: x = −1, 5
   • A1 x = −1, 0, 4, 5

4. (d) Find \(g(f(x))\). Give your answer in expanded form. [3]

   • M1 for substituting f(x) into g
   • M1 for expanding the squares or the product
   • A1 x^4 − 8x^3 + 11x^2 + 20x

5. (e) Write down the range of f. [1]

   • A1 f(x) ≥ −4

The graph of \(y=f(x)\), where \(f(x)=x^{2} - 6 x + 4\), has a minimum point at \((3, -5)\). [13]

1. (a) Write down the coordinates of the minimum point of the graph of (i) \(y=f(x)-2\), (ii) \(y=f(x+5)\), (iii) \(y=3f(x)\). [3]

   • A1 (i) (3, −7)
   • A1 (ii) (−2, −5)
   • A1 (iii) (3, −15)

2. (b) Find the coordinates of the minimum point of the graph of \(y=4f(3x)\). [1]

   • A1 (1, −20)

3. (c) The graph of \(y=x^2\) is transformed to the graph of \(y=2 x^{2} + 8 x + 3\). By completing the square, describe the two transformations. [4]

   • M1 for completing the square
   • A1 y = 2(x + 2)² − 5
   • A1 vertical stretch, scale factor 2
   • A1 translation: 2 units left and 5 units down

4. (d) The graph of \(y=f(x)\) is transformed to the graph of \(y=f(3x-6)\). Describe a sequence of two transformations that does this. [3]

   • M1 for a horizontal stretch with scale factor 1/3
   • M1 for a horizontal translation of the right size for the order used
   • A1 for the order matched to the size: translation 6 right, then horizontal stretch with scale factor 1/3; or horizontal stretch with scale factor 1/3, then translation 2 right

5. (e) Find the equation of the graph of \(y=f(x)\) after a reflection in the x-axis followed by a reflection in the y-axis. Give your answer in the form \(y=ax^2+bx+c\). [2]

   • M1 for y = −f(−x)
   • A1 y = −x^2 − 6x − 4

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

The graph of \(f(x)=\dfrac{8x}{x-3}\), for \(x>3\), has asymptotes that meet at the point \(Q(3, 8)\). The point \(P(x,\,y)\) lies on the graph of f. Use your calculator where it helps. [7]

1. (a) Show that \(PQ=\sqrt{(x-3)^2+\left(\dfrac{24}{x-3}\right)^2}\). [3]

   • M1 for PQ² = (x − 3)² + (y − 8)²
   • M1 for y − 8 = 8x/(x − 3) − 8 = 24/(x − 3)
   • A1 PQ = √((x − 3)² + (24/(x − 3))²)

2. (b) Find the shortest distance from P to Q. Give your answer to 3 significant figures. [2]

   • M1 for graphing PQ as a function of x and finding its minimum
   • A1 6.93

3. (c) Find the coordinates of P when PQ is least. Give each coordinate to 3 significant figures. [2]

   • A1 x = 7.90
   • A1 y = 12.9 (8x/(x − 3))

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

The function f is defined by \(f(x)=\ln x+3\), for \(x>0\), and the function g is defined by \(g(x)=(x+2)^2-4\), for \(x\ge -2\). [10]

1. (a) Find \(f^{-1}(x)\), and write down its domain and its range. [3]

   • M1 for y = ln x + 3, rearranged to x = e^(y − 3)
   • A1 f⁻¹(x) = e^(x − 3)
   • A1 domain: all real x; range: f⁻¹(x) > 0 (the range of f is all real values; the domain of f is x > 0)

2. (b) Find \(g^{-1}(x)\), and write down its domain and its range. [4]

   • M1 for x = (y + 2)² − 4 with x and y swapped, and a square root taken
   • A1 g⁻¹(x) = −2 + √(x + 4) (the positive root, as g⁻¹ has range x ≥ −2)
   • A1 domain x ≥ −4 (the range of g)
   • A1 range g⁻¹(x) ≥ −2 (the domain of g)

3. (c) Solve \(f(x)=f^{-1}(x)\). Give your answers to 3 significant figures. [3]

   • M1 for f(x) = f⁻¹(x) giving f(x) = x, since a function and its inverse meet on y = x
   • M1 for solving ln x + 3 = x with the calculator
   • A1 x = 0.0525 or x = 4.51