Functions, modulus and partial fractions – Year 13 A level maths worksheet with answers to download as PDF

Find the range of each function. State clearly any restriction on the domain that you use.

1. \(f(x)=x^2-6x+10\), \(x\in\mathbb{R}\) [3 marks]
2. \(g(x)=\sqrt{2x-4}\), with the largest possible real domain [3 marks]
3. \(h(x)=\dfrac3{x-2}\), \(x\ne2\) [2 marks]

The functions \(f\) and \(g\) are defined for all real \(x\) by \(f(x)=3x-1\) and \(g(x)=x^2+2\).

1. Find \(fg(x)\) and \(gf(x)\). [3 marks]
2. Solve \(fg(x)=gf(x)\), giving your answers in exact form. [4 marks]
3. Find the value of \(ff(2)\). [1 mark]

Find the inverse function, stating its domain, for each of the following.

1. \(f(x)=\dfrac{2x+1}{x-3}\), \(x\ne3\) [4 marks]
2. \(g(x)=e^{2x}-1\), \(x\in\mathbb{R}\) [3 marks]
3. \(h(x)=x^2+4\), \(x\ge0\) [3 marks]

Solve each equation.

1. \(|3x-1|=8\) [3 marks]
2. \(|x+2|=3x-2\) [4 marks]
3. \(|2x-1|=|x+4|\) [4 marks]

Solve each inequality.

1. \(|2x-5|<7\) [3 marks]
2. \(|x+3|\ge4\) [3 marks]
3. \(|x-1|<2x\) [5 marks]

Express each of the following in partial fractions.

1. \(\dfrac{7x+1}{(x+1)(x-2)}\) [3 marks]
2. \(\dfrac{4x-1}{(x-1)^2}\) [3 marks]
3. \(\dfrac{x^2+1}{x(x+1)}\) [5 marks]

A café in Manchester serves coffee at 90 °C. The temperature in °C after \(t\) minutes is modelled by \(T(t)=22+68e^{-0.08t}\), \(t\ge0\).

1. State the room temperature suggested by the model and the range of \(T\). [2 marks]
2. Find the inverse function \(T^{-1}\) and state its domain. [3 marks]
3. Find, to 3 significant figures, when the coffee has cooled to 60 °C. [2 marks]
4. A customer likes to drink at 50 °C. Find the time from serving. [2 marks]

A traveller changes pounds into euros. The bureau takes a 2% fee first, so the amount after the fee is \(f(x)=0.98x\). The remaining pounds are converted at €1.17 per £, so \(g(y)=1.17y\).

1. Find \(gf(x)\) and explain what it represents. [2 marks]
2. How many euros does £250 give? [1 mark]
3. Find the inverse function \((gf)^{-1}\) and use it to find how many pounds must be changed to receive €500, to the nearest penny. [3 marks]
4. Explain why the order of the two operations matters when the fee is a flat amount. [1 mark]

A factory fills bags of flour. A bag is accepted if its mass \(m\) grams satisfies \(|m-500|\le8\). The quality-control team weighed six bags.

1. Write the acceptance condition without a modulus sign. [1 mark]
2. Which bags are accepted? [2 marks]
3. The scale reads 2.5 g too high. State the accepted range of true masses if the reading must satisfy the original condition. [3 marks]
4. Sketch the graph of \(y=|m-500|\) and mark the region where bags are accepted. [2 marks]

Ana and Ben cycle towards each other along a straight 30 m practice track in a park. At time \(t\) seconds, Ana is \(2t\) metres from the west end and Ben is \(30-3t\) metres from the west end, for \(0\le t\le10\).

1. Show that the distance between them is \(d=|30-5t|\). [2 marks]
2. When are they exactly 10 m apart? [3 marks]
3. For how long are they within 5 m of each other? [3 marks]
4. When do they pass each other? [1 mark]

In photography, an object at distance \(u\) cm from a lens of focal length 10 cm forms a sharp image at distance \(v\) cm, where \(\dfrac1u+\dfrac1v=\dfrac1{10}\) and \(u>10\).

1. Show that \(v=g(u)=\dfrac{10u}{u-10}\). [2 marks]
2. Find the image distance for \(u=15\), \(20\) and \(30\). [2 marks]
3. Show that \(gg(u)=u\), and explain what this means for the inverse function. [3 marks]
4. State the range of \(g\). [2 marks]

Maya pays £200 into a savings account at the end of each year. The account adds 3% interest during the year. If the balance at the start of a year is £\(x\), the balance at the end is \(f(x)=1.03x+200\).

1. Find \(ff(x)\) and interpret it. [3 marks]
2. Find \(f^{-1}(y)\). [2 marks]
3. The balance after two years is £5000. Find the balance at the start of year 2, and at the start of year 1. [3 marks]

A school club orders \(n\) printed T-shirts. There is a set-up charge of £500 and each T-shirt costs £20 to print. The average cost per T-shirt is \(C(n)=\dfrac{500+20n}{n}\), \(n\ge1\).

1. Show that \(C(n)=20+\dfrac{500}n\) and find \(C(100)\). [2 marks]
2. The club wants an average cost of £22. How many T-shirts are needed? [2 marks]
3. Find the inverse function \(C^{-1}\) and state its domain. [3 marks]
4. Explain why the average cost can never fall below £20. [1 mark]

Two pumps fill a leisure-centre pool. Working together they fill it in 4 hours. The slower pump alone would take 6 hours longer than the faster one. Let the faster pump take \(a\) hours alone.

1. Show that \(\dfrac1a+\dfrac1{a+6}=\dfrac14\) and that this simplifies to \(a^2-2a-24=0\). [3 marks]
2. Solve to find how long each pump takes alone. [2 marks]
3. Show that \(\dfrac6{a(a+6)}=\dfrac1a-\dfrac1{a+6}\). [2 marks]
4. Find the fraction of the pool filled by both pumps together in 1 hour and check that it equals the answer from part (a). [2 marks]

The function \(f\) is defined by \(f(x)=|x-3|-2\), \(x\in\mathbb{R}\).

1. Sketch the graph of \(y=f(x)\), labelling the vertex and all intercepts. [3 marks]
2. State the range of \(f\). [1 mark]
3. Solve the equation \(f(x)=\dfrac x2\). [4 marks]
4. The function \(g\) is \(f(x)\) restricted to \(x\ge3\). Find \(g^{-1}(x)\) and state its domain. [3 marks]
5. Find the values of \(x\) for which \(f(x)<1\). [2 marks]

The functions \(f\) and \(g\) are defined by \(f(x)=2x-3\), \(x\in\mathbb{R}\) and \(g(x)=x^2+1\), \(x\ge0\).

1. Find \(fg(x)\) and state its range. [3 marks]
2. Explain why \(gf(x)\) only exists for \(x\ge\tfrac32\) and write down \(gf(x)\). [3 marks]
3. Solve \(fg(x)=gf(x)\), giving answers to 3 decimal places. [4 marks]
4. Find \(g^{-1}(x)\) and state its domain and range. [3 marks]

\(\(F(x)=\dfrac{3x^2+9x-3}{(x-1)(x+2)^2},\qquad x>1.\)\)

1. Find constants \(A\), \(B\) and \(C\) such that \(F(x)=\dfrac A{x-1}+\dfrac B{x+2}+\dfrac C{(x+2)^2}\). [5 marks]
2. Check your result by evaluating both forms at \(x=2\). [2 marks]
3. Describe the behaviour of \(F(x)\) as \(x\to\infty\). [2 marks]
4. Show that \(F(x)>0\) for all \(x>1\). [3 marks]

A machine cuts piston rings to length \(L\) mm. Rings are accepted if \(|L-120|\le0.4\). An inspector records the total cost of waste as \(5|L-120|\) pounds per ring, because rings further from 120 mm need more rework.

1. Find the accepted range of \(L\) and decide which rings are accepted. [3 marks]
2. Find the rework cost for each ring. [3 marks]
3. The company will only accept rework costs below £1.50. Solve \(5|L-120|<1.5\). [3 marks]
4. The gauge reads 0.15 mm too high. Find the true accepted range of lengths when the readings must satisfy \(|r-120|\le0.4\). [3 marks]

The number of grey seals on an island is modelled by \(P(t)=\dfrac{800}{1+3e^{-0.4t}}\), where \(t\) is the time in years after monitoring began.

1. Verify the value in the table at \(t=5\) and state the long-term population predicted by the model. [3 marks]
2. State the range of \(P\) for \(t\ge0\). [2 marks]
3. Find the inverse function \(t=P^{-1}(N)\). [4 marks]
4. Use your inverse to find when the population first reaches 600. [2 marks]
5. Comment on whether the model is sensible. [2 marks]

The function \(f\) is defined by \(f(x)=\dfrac{3x+k}{x-3}\), \(x\in\mathbb{R}\), \(x\ne3\), where \(k\) is a constant.

1. Show that \(ff(x)=x\) for every \(k\), and explain what this means. [4 marks]
2. Hence state \(f^{-1}(x)\). [1 mark]
3. Show that \(f(x)=x\) leads to \(x^2-6x-k=0\). [2 marks]
4. Find the range of values of \(k\) for which \(f(x)=x\) has two distinct real roots. [2 marks]
5. For \(k=7\) find the exact solutions of \(f(x)=f^{-1}(x)\). [2 marks]