Algebra & Functions

This problem sheet is split into four sections, split across algebra recap and new content. However, there are only four questions per topic. If there’s any particular question you find difficult, try using ELM to create more questions as a study aid.

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An example prompt might look like

Can you create 5 examples of questions similar to “What does 5a + 3b + 2a + b + 5c simplify to?” using a variety of phrasing and presentation technique.

As we’ve seen, generative AI isn’t always great at solving the problems it creates! Instead check your solutions to these problems using a tool like Wolfram Alpha, Desmos, or even just a scientific calculator.

1 Simplifying Expressions

These questions practice simplifying algebraic expressions.

  1. What does \(5a + 3b + 2a + b + 5c\) simplify to?

  2. What does \(x^2 - 2x + 3x^2 + 5x\) simplify to?

  3. What does \(n^2 \times n^3\) simplify to?

  4. What does \(\frac{p^8}{p^4}\) simplify to?

2 Linear Equations

  1. What is the solution to \(5x + 3 = 0\)?

  2. At what point does \(y = 0.2x - 0.5\) cross the \(y\)-axis? How about the \(x\)-axis?

  3. When does the lines \(y = 2x - 1\) and \(y = -2x + 3\) intersect?

  4. Explain in words the gradient \(m\) of a straight line \(y = mx + c\).

3 Quadratic Equations

  1. What is \(n\) if \(n^2 = 9\)?

  2. What are the roots of \(3y^2 + 5y - 1\)?

  3. At what points do the curves \(y = x^2 + 2x\) and \(7y + x^2 = 4\) intersect?

  4. What are the solutions to \(3x^2 - 48 = 0\)?

4 Functions

  1. What are the domain and range of \(f(x) = \sqrt{x}\)?

  2. If \(f(x) = x(x+1)\), what is the inverse function?

  3. Find functions \(f(x)\) and \(g(x)\) that prove \(f\circ g\neq g\circ f\).

  4. If \(h(x) = \sqrt{x^2 + 2}\), what is a possible decomposition of \(h\)? Are there others?