Chapter 15: Review

Return to the two machines

The first machine is $f(x)=2x+3$. The second is $g(x)=\frac{x-3}{2}$. For input $4$, the first then second gives

The second then first also gives

These two rules are inverses, so each undoes the other. In general, $g(f(x))=x$ and $f(g(x))=x$.

Mixed practice

Choose a method before working. Say why it fits, then check your result.

1. Find the inverse of $h(x)=5x-7$. Check by composing in both orders.
2. Simplify $\frac{x^2-9}{x^2+6x+9}$ and state excluded values.
3. Work out $\frac{2}{x}+\frac{3}{x+1}$, giving a simplified answer.
4. Expand $(x-4)(x+2)$, then factorise $x^2-2x-8$. Explain how the results connect.
5. Prove that the sum of three consecutive integers is divisible by $3$.
6. Chapter 12 revisited: solve $x^2-5x+6=0$. Which method applies, and why?
7. Chapter 14 revisited: solve $y=2x+1$ and $y=x+4$ simultaneously. Which method applies, and why?

Reflect

Which method can you now choose with more confidence? Name one question where you will check your answer by using an inverse operation or an equivalent form.