Functions and graph transformations – Year 11 GCSE maths worksheet with answers to download as PDF

The function \(f\) is defined by \(f(x)=3x-2\) and the function \(g\) by \(g(x)=x^2+1\).

1. Work out \(f(4)\), \(f(-1)\) and \(f(0.5)\). [3 marks]
2. Solve \(f(x)=19\). [2 marks]
3. Work out \(g(3)\) and \(g(-2)\). [2 marks]
4. Solve \(g(x)=26\). [2 marks]

\(f(x)=2x+3\) and \(g(x)=x^2\).

1. Find \(fg(x)\) and \(gf(x)\), and simplify each. [4 marks]
2. Work out \(fg(2)\) and \(gf(2)\). [2 marks]
3. Solve \(fg(x)=35\). [3 marks]
4. Explain why \(fg(x)\ne gf(x)\). [1 mark]

Find the inverse of each function.

1. \(f(x)=4x-7\) [2 marks]
2. \(g(x)=\dfrac{x+2}{5}\) [2 marks]
3. \(h(x)=\dfrac x3+1\) [2 marks]
4. Use your answer to (a) to find \(f^{-1}(9)\) and check that \(f(f^{-1}(9))=9\). [2 marks]

The domain of a function is the set of allowed inputs and the range is the set of outputs.

1. \(f(x)=x^2\) with domain \(-3\le x\le2\). Find the range. [3 marks]
2. \(g(x)=\dfrac1x\). Which value of \(x\) must be excluded from the domain? [1 mark]
3. \(h(x)=\sqrt{x-1}\). Write the domain and the range. [3 marks]
4. \(k(x)=2x+1\) with domain \(x\ge0\). What is the range? [2 marks]

The point \((3,4)\) is on the graph of \(y=f(x)\). Write down the coordinates of the image of this point on each graph.

1. \(y=f(x)+2\) [1 mark]
2. \(y=f(x+1)\) [2 marks]
3. \(y=2f(x)\) [1 mark]
4. \(y=f(2x)\) [2 marks]
5. \(y=-f(x)\) [1 mark]
6. \(y=f(-x)\) [1 mark]

Sketch each graph, marking where it crosses the axes and any turning point.

1. \(y=x^2-4x+3\) [4 marks]
2. \(y=2^x\) [3 marks]
3. \(y=\dfrac1x\) [2 marks]

A taxi company charges £2.50 plus £1.60 per mile. The fare for \(d\) miles is \(f(d)=2.5+1.6d\).

1. Work out \(f(5)\). [1 mark]
2. Find the inverse function \(f^{-1}\) and explain what it tells you. [3 marks]
3. Use it to find how far you can travel for £15.70. [2 marks]
4. The company adds a 10% tip. Write the total as a composite function \(g(f(d))\) where \(g(x)=1.1x\), and find the total for 5 miles. [3 marks]

A bank charges a fixed fee of £3 and then converts pounds to euros at a rate of 1.15 euros per pound. Let \(h(x)=x-3\) be the amount after the fee and \(g(x)=1.15x\) the conversion.

1. Write \(gh(x)\) and explain what it represents. [2 marks]
2. How many euros do you get for £200? [2 marks]
3. Find \((gh)^{-1}(x)\). [3 marks]
4. Use the inverse to find how many pounds you need to get 345 euros. [2 marks]

The temperature in degrees Celsius is \(C(F)=\dfrac59(F-32)\) where \(F\) is the temperature in degrees Fahrenheit.

1. Work out \(C(77)\). [2 marks]
2. Find the inverse function \(C^{-1}\). [3 marks]
3. Use it to find the Fahrenheit temperature that equals 37°C. [2 marks]
4. Show that the two scales agree at \(-40\). [2 marks]

A shop gives a 10% discount, \(f(x)=0.9x\), and then adds 20% VAT, \(g(x)=1.2x\). \(x\) is the price in pounds.

1. Write \(gf(x)\) and simplify. [2 marks]
2. Work out the final price of a laptop with starting price £250. [1 mark]
3. Does it matter whether the VAT is added before or after the discount? Show that \(fg(x)=gf(x)\). [3 marks]
4. Find \((gf)^{-1}(x)\) and use it to find the starting price when the final price is £162. [4 marks]

A ball is dropped from a height of 20 m. Its height after \(t\) seconds is \(h(t)=20-5t^2\) for \(0\le t\le2\).

1. Work out \(h(0)\), \(h(1)\) and \(h(2)\). [3 marks]
2. Write the range of \(h\). [2 marks]
3. Solve \(h(t)=10\) to 2 decimal places. [3 marks]
4. Explain why the domain stops at \(t=2\). [1 mark]

The graph of \(y=f(x)\) has its minimum point at \((2,-3)\).

1. Write the minimum point of \(y=f(x+4)\). [2 marks]
2. Write the minimum point of \(y=f(x)-5\). [2 marks]
3. Write the minimum point of \(y=2f(x)\). [2 marks]
4. Write the minimum point of \(y=f(2x)\). [2 marks]
5. The graph of \(y=-f(x)\) has a turning point. Is it a maximum or a minimum? Give its coordinates. [2 marks]

The graph of \(y=2^x\) is drawn for \(-1\le x\le3\).

1. Complete a table of values for \(x=-1,0,1,2,3\). [2 marks]
2. The graph of \(y=2^x+3\) is drawn. Write the equation of the horizontal asymptote and the \(y\)-intercept. [3 marks]
3. Describe the transformation that maps \(y=2^x\) to \(y=2^{x+1}\). [2 marks]
4. Show that \(y=2^{x+1}\) is the same as \(y=2\times2^x\) and explain what this means for the graph. [3 marks]

Plan A costs £15 joining fee plus £8 per month: \(a(m)=15+8m\). Plan B costs £30 joining fee plus £5 per month: \(b(m)=30+5m\).

1. Solve \(a(m)=b(m)\). [3 marks]
2. What is the cost of each plan at that time? [1 mark]
3. Find \(a^{-1}(x)\) and use it to find how many months you can pay for with £111 on plan A. [3 marks]
4. Which plan is cheaper over one year? [2 marks]

\(f(x)=3x-1\) and \(g(x)=x^2+2\).

1. Find \(fg(x)\) and \(gf(x)\). Expand and simplify. [4 marks]
2. Solve \(fg(x)=53\). [3 marks]
3. Show that the equation \(fg(x)=gf(x)\) simplifies to \(3x^2-3x-1=0\). [3 marks]
4. Solve it to 2 decimal places. [3 marks]

\(f(x)=\dfrac{2x+1}{x-3}\) for \(x\ne3\).

1. Work out \(f(4)\). [1 mark]
2. Show that \(f^{-1}(x)=\dfrac{3x+1}{x-2}\). [5 marks]
3. Check that \(f^{-1}(9)=4\). [2 marks]
4. State a value of \(x\) that must be excluded from the domain of \(f^{-1}\). [1 mark]

\(f(x)=x^2\). The graph of \(y=g(x)\) is the graph of \(y=f(x+2)-5\).

1. Write \(g(x)\) in the form \((x+a)^2+b\) and state the turning point. [3 marks]
2. Describe the transformation that maps \(y=f(x)\) to \(y=g(x)\). [2 marks]
3. Find the points where the graph of \(g\) crosses the \(x\)-axis, to 2 decimal places. [4 marks]
4. Write the range of \(g\). [1 mark]

\(f(x)=x+5\) and \(fg(x)=3x+2\).

1. Find \(g(x)\). [3 marks]
2. Find \(gf(x)\). [2 marks]
3. Find \(g^{-1}(x)\). [2 marks]
4. Solve \(g(x)=g^{-1}(x)\). [3 marks]

The graph of \(y=f(x)\) passes through \((0,0)\), \((2,4)\) and \((4,0)\).

1. Complete the table. [5 marks]
2. Which of these graphs has the same maximum height as \(y=f(x)\)? [1 mark]
3. Which graph has a maximum point at \((1,4)\)? [1 mark]

\(f(x)=ax+b\). \(f(1)=5\) and \(ff(1)=17\).

1. Use \(f(1)=5\) to find the value of \(f(5)\) and write two equations for \(a\) and \(b\). [3 marks]
2. Solve them to find \(a\) and \(b\). [3 marks]
3. Find \(f^{-1}(x)\). [2 marks]
4. Solve \(f(x)=f^{-1}(x)\). [3 marks]