Functions: exam review Exam review: second paper

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]
2. (b) Find \(g(f(x))\). Give your answer in expanded form. [2]
3. (c) Find the largest possible domain of \(h(f(x))\). [2]

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]
2. (b) The point A is reflected in the line \(y=x\). Write down the coordinates of its image. [1]
3. (c) Write down the value of \(g(f(1.8))\). [1]
4. (d) Write down how the functions f and g are related. [1]

The function f is defined by \(f(x)=\dfrac{7x-4}{2}\). [5]

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

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]
2. (b) Write down the range of f for the domain \(x\ge k\). [1]
3. (c) Find \(f^{-1}(-13)\). [2]

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

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

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]
2. (b) Say what the horizontal asymptote tells you about the number of subscribers. [1]
3. (c) Find the number of subscribers after 10 weeks. [1]

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

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]
2. (b) Solve \(f(g(x))=32\). [2]
3. (c) Solve \(f(g(x))=g(f(x))\). Give your answers to 3 significant figures. [3]

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]
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]
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]

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]
2. (b) Use the graphs to find \(g(f(5))\). [2]
3. (c) Solve \(g(f(x))=0\). [4]
4. (d) Find \(g(f(x))\). Give your answer in expanded form. [3]
5. (e) Write down the range of f. [1]

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

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]
2. (b) Find b. [2]
3. (c) Given that \(D(1)=8\), find the value of k. [3]
4. (d) Find the distance still to travel after 4 hours. [2]
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]

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]
2. (b) Find the shortest distance from P to Q. Give your answer to 3 significant figures. [2]
3. (c) Find the coordinates of P when PQ is least. Give each coordinate to 3 significant figures. [2]

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]
2. (b) Find the value of b. [1]
3. (c) Find the value of k. [2]
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]
5. (e) Use your calculator to find the time, to the nearest hour, when the ferry has travelled 39.5 km. [2]

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]
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]
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]
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]
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]

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]
2. (b) Find \(g^{-1}(x)\), and write down its domain and its range. [4]
3. (c) Solve \(f(x)=f^{-1}(x)\). Give your answers to 3 significant figures. [3]