Functions: exam review Exam review One question at a time

The functions f, g and h are defined by \(f(x)=3 x + 2\), \(g(x)=x^{2} - 1\) and \(h(x)=\sqrt{x - 4}\). [6]

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

   • M1 for substituting g(x) into f
   • A1 3x^2 − 1

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

   • M1 for substituting f(x) into g
   • A1 9x^2 + 12x + 3

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

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

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 2, 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 (2, 7.39)

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

   • A1 (7.39, 2)

3. (c) Write down the value of \(g(f(2))\). [1]

   • A1 2 (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 \(f(x)=\dfrac{5x+2}{3}\). [5]

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

   • M1 for swapping x and y and making y the subject: x = (5y + 2)/3
   • A1 f⁻¹(x) = (3x − 2)/5

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

   • M1 for substituting f⁻¹(x) into f
   • A1 f(f⁻¹(x)) = (5×(3x − 2)/5 + 2)/3
   • A1 = 3x/3 = x

The function f is defined by \(f(x)=2 x^{3} - 7 x^{2} + 4\). 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 = 2.33

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

   • A1 f(x) ≥ −8.70

3. (c) Find \(f^{-1}(-5)\). [2]

   • M1 for solving f(x) = −5 for x ≥ k
   • A1 3

The function f is defined by \(f(x)=\dfrac{5x}{x-2}\), \(x\ne 2\). [5]

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

   • A1 x = 2
   • A1 y = 5

2. (b) The graph of \(g(x)=\dfrac{bx}{x-c}\) has asymptotes that meet at the point \((4, 3)\). Find b and c. [2]

   • A1 c = 4 (vertical asymptote x = c)
   • A1 b = 3 (horizontal asymptote y = b)

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

   • A1 f(x) ≠ 5

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 0.5, lies on the graph of f. [4]

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

   • A1 (0.5, 3.16) (\(10^{0.5}=3.16\))

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(3.16, 0.5)
   • A1 y = x

3. (c) Write down the value of \(g(3.16)\). Give your answer to 3 significant figures. [1]

   • A1 0.500 (B is on the graph of g, so \(g(3.16)=0.5\))

The number of members N of a club, t months after it opens, is modelled by \(N(t)=\dfrac{450t}{t+3}\), for \(t\ge0\). [4]

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

   • A1 t = −3
   • A1 N = 450

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

   • A1 N tends to 450 as t grows, but never reaches it

3. (c) Find the number of members after 6 months. [1]

   • A1 300 (\(450\times6\div9\))

A photo shop prints holiday photos. The cost \(C\) euros of an order of n photos is modelled by \(C(n)=2.5+0.35n\). [9]

1. (a) Find \(C(40)\), and find the value of n for which \(C(n)=23.5\). [3]

   • A1 C(40) = 16.50
   • M1 for 2.5 + 0.35n = 23.5
   • A1 n = 60

2. (b) The shop accepts at most 120 photos in one order. Write down a reasonable domain and a reasonable range for this model. [2]

   • A1 domain 0 ≤ n ≤ 120, n a whole number (you cannot order part of an item)
   • A1 range 2.5 ≤ C ≤ 44.50

3. (c) The function g is defined by \(g(x)=\sqrt{x-4}+3\). Write down the domain and the range of g. [2]

   • A1 x ≥ 4 (the number under the square root cannot be negative)
   • A1 g(x) ≥ 3 (a square root is at least 0)

4. (d) The function h is defined by \(h(x)=x^{2} - 6 x + 5\), for \(0\le x\le 5\). 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 −4 ≤ h(x) ≤ 5

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

1. (a) The function h is defined by \(h(x)=\dfrac{1}{x^{2}+3}\). Write h in the form \(h(x)=p(q(x))\), and give the functions p and q. [2]

   • A1 q(x) = x² + 3 (the inside function)
   • A1 p(x) = 1/x (the outside function)

2. (b) Solve \(f(g(x))=19\). [2]

   • M1 for f(g(x)) = 2x² + 1
   • 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)) = 4x² + 12x + 8 and f(g(x)) = g(f(x))
   • M1 for solving the quadratic equation (calculator or formula)
   • A1 x = −5.35 or x = −0.655

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

1. (a) The graph of \(y=x^2+2x\) is translated 3 units to the right. Find the equation of the new graph in the form \(y=ax^2+bx+c\). [2]

   • M1 for replacing x by (x − 3) in y = x² + 2x
   • A1 y = x² − 4x + 3

2. (b) The graph of \(y=3^x\) is stretched vertically with scale factor 4, and then translated 5 units down. Write down the equation of the new graph and the equation of its asymptote. [3]

   • M1 for stretching first: 4(3^x)
   • A1 y = 4(3^x) − 5
   • A1 asymptote y = −5 (the asymptote y = 0 is stretched to y = 0 and then moved 5 down)

3. (c) The point \((4, -6)\) lies on the graph of \(y=f(x)\). The graph of \(y=-3f(-2x)\) 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 3, with a reflection in the x-axis
   • A1 horizontal stretch, scale factor 1/2, with a reflection in the y-axis
   • M1 for x-coordinate 4 ÷ (−2) and y-coordinate (−3) × (−6)
   • A1 (−2, 18)

The functions f and g are defined by \(f(x)=x^{2} - 4 x + 3\) and \(g(x)=x^{2} - 3 x\). Their graphs are shown. [12]

1. (a) Use the graphs to find \(f(g(4))\). [2]

   • M1 for reading g(4) = 4 from the graph of g
   • A1 f(4) = 3

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

   • M1 for reading f(2) = -1 from the graph of f
   • A1 g(-1) = 4

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

   • M1 for g(u) = 0 where u = f(x): u = 0 or 3
   • M1 for solving f(x) = 0: x = 1, 3
   • M1 for solving f(x) = 3: x = 0, 4
   • A1 x = 0, 1, 3, 4

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 + 19x^2 − 12x

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

   • A1 f(x) ≥ −1

The graph of \(y=f(x)\), where \(f(x)=-x^{2} + 4 x + 1\), has a maximum point at \((2, 5)\). [13]

1. (a) Write down the coordinates of the maximum point of the graph of (i) \(y=f(x)+3\), (ii) \(y=f(x-4)\), (iii) \(y=-2f(x)\). [3]

   • A1 (i) (2, 8)
   • A1 (ii) (6, 5)
   • A1 (iii) (2, −10)

2. (b) Find the coordinates of the maximum point of the graph of \(y=2f(\frac{1}{2}x)\). [1]

   • A1 (4, 10)

3. (c) The graph of \(y=x^2\) is transformed to the graph of \(y=3 x^{2} - 12 x + 7\). By completing the square, describe the two transformations. [4]

   • M1 for completing the square
   • A1 y = 3(x − 2)² − 5
   • A1 vertical stretch, scale factor 3
   • A1 translation: 2 units right and 5 units down

4. (d) The graph of \(y=f(x)\) is transformed to the graph of \(y=f(2x+6)\). Describe a sequence of two transformations that does this. [3]

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

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 + 4x − 1

A boat travels towards a harbour that is 18 km from where it starts. Its distance D km from the start after t hours is modelled by \(D=a+bk^{-t}\), where \(k>1\). [10]

1. (a) Explain why a is equal to 18. [1]

   • R1 as t gets large, k^(−t) tends to 0, so D approaches a, the whole 18 km

2. (b) Find b. [2]

   • M1 for D(0) = 0, so a + b = 0
   • A1 b = −18

3. (c) Given that \(D(1)=5.4\), find the value of k. [3]

   • M1 for 18 − 18k^(−1) = 5.4
   • M1 for k^(−1) = 0.700 (12.6/18)
   • A1 k = 1.43

4. (d) Find the distance still to travel after 3 hours. [2]

   • M1 for 18 − D(3) or 18k^(−3)
   • A1 6.17

5. (e) Find the time at which the boat is 1 km from the end. Give your answer in hours to 3 significant figures. [2]

   • M1 for solving 18 − D(t) = 1, or 18k^(−t) = 1
   • A1 8.10

The graph of \(f(x)=\dfrac{6x}{x-2}\), for \(x>2\), has asymptotes that meet at the point \(Q(2, 6)\). The point \(P(x,\,y)\) lies on the graph of f. Use your calculator where it helps. [7]

1. (a) Show that \(PQ=\sqrt{(x-2)^2+\left(\dfrac{12}{x-2}\right)^2}\). [3]

   • M1 for PQ² = (x − 2)² + (y − 6)²
   • M1 for y − 6 = 6x/(x − 2) − 6 = 12/(x − 2)
   • A1 PQ = √((x − 2)² + (12/(x − 2))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 4.90

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

   • A1 x = 5.46
   • A1 y = 9.46 (6x/(x − 2))

A freight train travels towards a depot that is 45 km from where it starts. Its distance D km from the start after t hours is modelled by \(D=a+bk^{-t}\), where \(k>1\). The train starts at \(D=0\), and after 1 hour it has travelled 20 km. [9]

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

   • R1 as t increases, \(k^{-t}\) tends to 0, so D tends to a
   • A1 a = 45

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

   • A1 b = −45 (\(D(0)=0\): \(45+b=0\))

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

   • M1 for \(45-45k^{-1}=20\)
   • A1 k = 1.80

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

   • M1 for \(45\times1.80^{-3}\) (the distance remaining is \(45-D\))
   • A1 7.72

5. (e) Use your calculator to find the time, to the nearest hour, when the train has travelled 44.5 km. [2]

   • M1 for solving \(45-45\times1.80^{-t}=44.5\)
   • A1 8 (t = 7.66, to the nearest hour)

A bakery uses four models in its planning. [16]

1. (a) The profit \(P\) euros from selling loaves at x euros each is modelled by \(P(x)=- 3 x^{2} + 36 x - 60\). 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 −60
   • A1 zeros x = 2 and x = 10
   • A1 vertex (6, 48)
   • R1 the greatest profit is 48 euros, made when the price is 6 euros

2. (b) The temperature of a loaf taken from the oven is modelled by \(T(t)=22+178e^{-0.06t}\), where T is in °C and t is in minutes after it leaves the oven. Write down the equation of the horizontal asymptote of the graph of T, say what it means in this context, and find T after 15 minutes. Give your answer to 3 significant figures. [3]

   • A1 T = 22
   • R1 the temperature of the loaf falls towards 22 °C (the temperature of the kitchen) but never goes below it
   • A1 94.4 °C

3. (c) The number N of loaves sold in a day is inversely proportional to the price p euros, so \(N=\dfrac{a}{p}\). When \(p=2.5\), \(N=80\). Find a, and find N when \(p=4\). Explain why the graph of N against p has the vertical axis as an asymptote. [4]

   • M1 for a = 80 × 2.5
   • A1 a = 200
   • A1 N = 50
   • 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 loaf is modelled by \(C(n)=\dfrac{4n+60}{n+5}\), 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 12 (C(0) = 60/5)
   • A1 vertical asymptote n = −5
   • A1 horizontal asymptote C = 4

5. (e) The amount y of dough after h hours is measured: when \(h=0, 1, 2\), y is 5, 10, 20. Say whether a linear, a quadratic or an exponential model fits better, and justify your choice. [2]

   • A1 exponential
   • R1 y is multiplied by 2 each hour (a constant ratio), not increased by a constant amount

The function f is defined by \(f(x)=\ln x+2\), for \(x>0\), and the function g is defined by \(g(x)=(x-3)^2+1\), for \(x\ge 3\). [10]

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

   • M1 for y = ln x + 2, rearranged to x = e^(y − 2)
   • A1 f⁻¹(x) = e^(x − 2)
   • 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 − 3)² + 1 with x and y swapped, and a square root taken
   • A1 g⁻¹(x) = 3 + √(x − 1) (the positive root, as g⁻¹ has range x ≥ 3)
   • A1 domain x ≥ 1 (the range of g)
   • A1 range g⁻¹(x) ≥ 3 (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 + 2 = x with the calculator
   • A1 x = 0.159 or x = 3.15