Building on composite functions
I can expand my thinking on composite functions to consider a more efficient way to write composite functions comprised of many functions.
- Year 11
Building on composite functions
I can expand my thinking on composite functions to consider a more efficient way to write composite functions comprised of many functions.
Lesson details
Key learning points
- A composite function could be many iterations of one original function
- To write this in function notation would be cumbersome
- 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.
Licence
Lesson video
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Prior knowledge starter quiz
6 Questions
Q1.__________ interest is when the interest is calculated on the original amount and the interest accumulated over the previous period.
Q2.Which of the below is the formula for compound interest?
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? £
Q4.If $$\text{f}(x)=13x-17$$ evaluate $$\text{f}(3)$$
Q5.$$\text{f}(x)=10-3x$$ Match the composite functions to their respective values.
$$\text{f}(4)$$ - $$-2$$
$$\text{ff}(4)$$ - $$16$$
$$\text{fff}(4)$$ - $$-38$$
$$\text{ffff}(4)$$ - $$124$$
$$\text{fffff}(4)$$ - $$-362$$
$$\text{ffffff}(4)$$ - $$1096$$
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 $$1$$'?
Assessment exit quiz
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.
Q2.In iteration notation how do we communicate 'Year 6'?
Q3.In iteration notation we use $$y_t$$ to express the current year. How do we express the next year?
Q4.$$£250$$ is invested at a rate of $$4$$% interest for $$3$$ years. Which iterative formula links the current term to the next term?
Q5.Match the iterative formulae to their matching scenario.
£500 invested, 2.5% interest - $$y_{t+1}=y_t\times1.025, \ y_0=500$$
£500 invested, 12.5% interest - $$y_{t+1}=y_t\times1.125, \ y_0=500$$
£250 invested, 2.5% interest - $$y_{t+1}=y_t\times1.025, \ y_0=250$$
£250 invested, 5% interest - $$y_{t+1}=y_t\times1.05, \ y_0=250$$
£500 invested, 5% interest - $$y_{t+1}=y_t\times1.05, \ y_0=500$$
Q6.If we were simplifying multiple representations of the function $$\text{f}(x)=3x^2-7x+1$$ which iterative formula would we use?
