Functions: exam review Exam review

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]
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 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]
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 \(g(f(2))\). [1]
4. (d) Write down how the functions f and g are related. [1]

The function f is defined by \(f(x)=\dfrac{5x+2}{3}\). [5]

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

The function f is defined by \(f(x)=2 x^{3} - 7 x^{2} + 4\). 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]
2. (b) Write down the range of f for the domain \(x\ge k\). [1]
3. (c) Find \(f^{-1}(-5)\). [2]

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]
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]
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 0.5, 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 \(g(3.16)\). Give your answer to 3 significant figures. [1]

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]
2. (b) Say what the horizontal asymptote tells you about the number of members. [1]
3. (c) Find the number of members after 6 months. [1]

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

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]
2. (b) Solve \(f(g(x))=19\). [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 \(y=x^2+2x\) is translated 3 units to the right. Find the equation of the new graph in the form \(y=ax^2+bx+c\). [2]
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]
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]

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]
2. (b) Use the graphs to find \(g(f(2))\). [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 \(f(x)=-x^{2} + 4 x + 1\), 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]
2. (b) Find the coordinates of the maximum point of the graph of \(y=2f(\frac{1}{2}x)\). [1]
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]
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]
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 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]
2. (b) Find b. [2]
3. (c) Given that \(D(1)=5.4\), find the value of k. [3]
4. (d) Find the distance still to travel after 3 hours. [2]
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]

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]
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 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]
2. (b) Find the value of b. [1]
3. (c) Find the value of k. [2]
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]
5. (e) Use your calculator to find the time, to the nearest hour, when the train has travelled 44.5 km. [2]

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]
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]
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]
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]
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]

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]
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]