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      Building on composite functions

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      Lesson details

      Learning outcome

      I can expand my thinking on composite functions to consider a more efficient way to write composite functions comprised of many functions.

      Key learning points

      1. A composite function could be many iterations of one original function
      2. To write this in function notation would be cumbersome
      3. There is more efficient notation that can be used in these circumstances

      Keywords

      • Iteration - Iteration is the repeated application of a function or process in which the output of each iteration is used as the input for the next iteration.

      Common misconception

      Pupils are very familiar with exponents but the introduction of 'subscript' notation may confuse them. For example, they may think $$x_2$$ must be a function of $$x$$ because $$x^2$$ is.

      Use of language helps. When talking through an iterative formula like $$x_{t+1}=8x_t +6$$ say, "The next $$x$$ value is equal to eight lots of the current $$x$$ value plus six" whilst pointing at it. Get pupils to do the same back to you.

      Teacher tip

      Use mini whiteboards to check pupils are grasping the notation. "How would we write 'The 10^{th} year'?" "If we use $$x_n$$ for the existing $$x$$ value what do we write for the next $$x$$ value?" Another useful question is "What is the difference between $$xn+1$$ and $$x_{n+1}$$?"

      Licence

      This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0
       except where otherwise stated. See Oak's terms & conditions
       (Collection 2).

      Lesson video

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      Prior knowledge starter quiz

      Download starter quiz questions (PDF)

      6 Questions

      Q1.
      __________ interest is when the interest is calculated on the original amount and the interest accumulated over the previous period.

      Simple
      Correct answer: Compound
      Investment

      Q2.
      Which of the below is the formula for compound interest?

      Correct answer: $$P\times(1+r)^t=A$$
      $$A\times(1+r)^t=P$$
      $$P\times(1\times{r})^t=A$$
      $$P\times(1+t)^r=A$$

      Q3.
      $$£2000$$ is invested at a rate of $$6.5$$% interest for $$3$$ years. How much is the investment worth at the end of the $$3$$ years? £

      Correct Answer: 2415.90

      Q4.
      If $$\text{f}(x)=13x-17$$ evaluate $$\text{f}(3)$$

      Correct Answer: 22, f(3)=22

      Q5.
      $$\text{f}(x)=10-3x$$ Match the composite functions to their respective values.

      Correct Answer:$$\text{f}(4)$$,$$-2$$

      $$-2$$

      Correct Answer:$$\text{ff}(4)$$,$$16$$

      $$16$$

      Correct Answer:$$\text{fff}(4)$$,$$-38$$

      $$-38$$

      Correct Answer:$$\text{ffff}(4)$$,$$124$$

      $$124$$

      Correct Answer:$$\text{fffff}(4)$$,$$-362$$

      $$-362$$

      Correct Answer:$$\text{ffffff}(4)$$,$$1096$$

      $$1096$$

      Q6.
      This table shows the yearly values when $$£800$$ is invested in a savings account with compound interest. What is the unknown value of 'Year $$1$$'?

      An image in a quiz
      Correct Answer: 880

      Assessment exit quiz

      Download exit quiz questions (PDF)

      6 Questions

      Q1.
      Iteration is the repeated application of a function or process in which the output of each iteration is used as the for the next iteration.

      Correct Answer: input

      Q2.
      In iteration notation how do we communicate 'Year 6'?

      $$y^6$$
      $$6y$$
      $$y6$$
      Correct answer: $$y_6$$

      Q3.
      In iteration notation we use $$y_t$$ to express the current year. How do we express the next year?

      $$y_t$$
      $$y_u$$
      $$y_t+1$$
      Correct answer: $$y_{t+1}$$

      Q4.
      $$£250$$ is invested at a rate of $$4$$% interest for $$3$$ years. Which iterative formula links the current term to the next term?

      $$y_{t}=y_{t+1}\times1.04$$
      Correct answer: $$y_{t+1}=y_t\times1.04$$
      $$y_{t+1}=y_t\times1.03$$
      $$y_{t}=y_t\times1.04$$

      Q5.
      Match the iterative formulae to their matching scenario.

      Correct Answer:£500 invested, 2.5% interest,$$y_{t+1}=y_t\times1.025, \ y_0=500$$

      $$y_{t+1}=y_t\times1.025, \ y_0=500$$

      Correct Answer:£500 invested, 12.5% interest,$$y_{t+1}=y_t\times1.125, \ y_0=500$$

      $$y_{t+1}=y_t\times1.125, \ y_0=500$$

      Correct Answer:£250 invested, 2.5% interest,$$y_{t+1}=y_t\times1.025, \ y_0=250$$

      $$y_{t+1}=y_t\times1.025, \ y_0=250$$

      Correct Answer:£250 invested, 5% interest,$$y_{t+1}=y_t\times1.05, \ y_0=250$$

      $$y_{t+1}=y_t\times1.05, \ y_0=250$$

      Correct Answer:£500 invested, 5% interest,$$y_{t+1}=y_t\times1.05, \ y_0=500$$

      $$y_{t+1}=y_t\times1.05, \ y_0=500$$

      Q6.
      If we were simplifying multiple representations of the function $$\text{f}(x)=3x^2-7x+1$$ which iterative formula would we use?

      Correct answer: $$x_{n+1}=3(x_n)^2-7(x_n)+1$$
      $$x_{n}=3(x_{n+1})^2-7(x_{n+1})+1$$
      $$x_{n+1}=3(x_n)^2-7x+1$$
      $$x_{n}=3x^2-7x+1$$

      To help you plan your 11 maths lesson on: Building on composite functions, download all teaching resources for free and adapt to suit your pupils' needs...