Understanding Piecewise Functions in Mathematics

Piecewise functions in mathematics are defined by multiple sub-functions, each applying to a specific interval of the main function's domain. These functions are often represented by different pieces or segments, and determining their ranges can involve analyzing various conditions and intervals. Check out the given content for detailed examples and exercises on working with piecewise functions.

• Mathematics
• Functions
• Piecewise
• Graphs
• Analysis

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1. Piecewise Functions

2. Piecewise Functions In mathematics, a piecewise-defined function (also called a piecewise function or a hybrid function) is a function defined by multiple sub-functions, each sub-function applying to a certain interval of the main function's domain, a sub-domain.

3. Piecewise Functions Sometimes functions are defined in pieces , with a different function for different ranges of ? values.

4. Piecewise Functions Sketch > Sketch > Sketch > (2, 9) (0, 5) (-1, 0) (5, 0)

5. Piecewise Functions ?2 1 0 ? < 1 1 ? < 2 2 ? < 3 Sketch > ? ? = Sketch > 3 ? Sketch > This example was used on the specification itself! (2, 1) (1, 1) (3, 0)

6. Range of Piecewise Functions The sketch shows the graph of ? = ?(?) with the domain 0 ? 9 3 0 ? < 2 2 ? < 4 4 ? 9 ? ? = ? + 1 9 ? Determine the range of ?(?). Range: ? ? ? ? Range ? Graph ?

7. Test Your Understanding The function ?(?) is defined for all ?: 4 ?2 ? < 2 2 ? 2 ? > 2 ? ? = 12 4? Determine the range of ?(?). Range: ? ? ? Range ? Graph ?

8. Test Your Understanding

10. Exercise Work out the range for each of these functions. (a) ? ? = ?2+ 6 for all ? ? ? ? (b) ? ? = 3? 5, 2 ? 6 ?? ? ? ?? (c) ? ? = 3?4, ? < 2 ? ? > ?? 4 [Set 1 Paper 2] (a) The function ?(?) is defined as: ? ? = 22 7?, The range of ?(?) is 13 ? ? 36 Work out the value of ?. ? = ? 1 2 ? ? ? ? ? (b) The function ?(?) is defined as ? ? = ?2 4? + 5 for all ?. (i) Express ?(?) in the form ? ?2+ ? ? ? = ? ??+ ? (ii) Hence write down the range of ?(?). ? ? ? ? ? (a) ? ? =?+2 Give a reason why ? > 0 is not a suitable domain for ?(?). It would include 3, for which ?(?) is undefined. (b) Give a possible domain for ? ? = ? 5? ? 2 ? 3 ? ? [June 2012 Paper 1] ? ? = 2?2+ 7 for all values of ?. (a) What is the value of ? 1 ? ? ? = ? (b) What is the range of ?(?)? ? ? ? 5 ? ? ? ? = 3 2?, The range of ?(?) is 5 < ? ? < 5 Work out ? and ?. ? = ?,? = ? ? < ? < ? 3 ? ?

11. Exercise [Jan 2013 Paper 2] ? ? = sin? 180 ? 360 ? ? = cos? 0 ? ? (a) What is the range of ?(?)? 6 8 ? ? ? ? ? (b) You are given that 0 ? ? 1. Work out the value of ?. ? = ?? ? Here is a sketch of ? ? = ?2+ 6? + ? for all ?, where ? is a constant. The range of ?(?) is ? ? 11. Work out the value of ?. ? ? = ? + ?? ? + ? ? + ? = ?? ? = ?? By completing the square or otherwise, determine the range of the following functions: (a) ? ? = ?2 2? + 5, for all ? = ? ??+ ? Range: ? ? ? (b) ? ? = ?2+ 6? 2, for all ? = ? + ?? ?? Range: ? ? ?? 7 ? ? ?

12. Exercise 9 10 ?(?) is a quadratic function with domain all real values of ?. Part of the graph of ? = ? ? is shown. (a) Write down the range of ?(?). ? ? ? (b) Use the graph to find solutions of the equation ? ? = 1. ? = ?.?,?.? (c) Use the graph to solve ? ? < 0. ? < ? ?? ? > ? The straight line shows a sketch of ? = ?(?) for the full domain of the function. (a) State the domain of the function. ? ? ? ?? (b) Work out the equation of the line. ? ? = ?? + ?? ? ? ? ? ?

13. Exercise The function ?(?) is defined as: 11 13 ? ? = ?2 4 0 ? < 3 3 ? 5 14 3? Work out the range of ? ? . ?(?) ? ? The function ?(?) has the domain 3 ? 3 and is defined as: 12 [June 2012 Paper 2] A sketch of ? = ?(?) for domain 0 ? 8 is shown. The graph is symmetrical about ? = 4. The range of ?(?) is 0 ? ? 12. Work out the function ?(?). ? ? = ? ? ? ? = ?2+ 3? + 2 3 ? < 0 0 ? 3 2 + ? Work out the range of ? ? . ? ? ? ? ? ? 0 ? 4 4 < ? 8 ?? ? ? ? ? < ? ? ? ? ? = ?? ??

14. Constructing a function June 2013 Paper 2 What would be the simplest function to use that has this domain/range? A straight line! Note, that could either be going up or down (provided it starts and ends at a corner) ? 11 ? What is the equation of this? ? =? ?= ? 3 ? ? = ? ? ? ? = ?? + ? ? ? = ?? + ? ? ? 5 1

15. Constructing a function Sometimes there s the additional constraint that the function is increasing or decreasing . ? ? is a decreasing function with domain 4 ? 6 and range 7 ? ? 19. ? ? = ?? ?= ? 19 ? ? = ? ? ? ? = ?? + ?? ? ? = ?? ?? Function 7 Graph ? 6 4

16. Exercise Domain is 1 ? < 3. Range 1 ? ? 3. ?(?) is an increasing function. ? ? = ? 1 ? Domain is 1 ? 3. Range 1 ? ? 3. ?(?) is a decreasing function. ? ? = ?? ? 2 ? Domain is 5 ? 7. Range 7 ? ? 11. ?(?) is an increasing function. ? ? = ??? ? 3 ? 4 Domain is 5 ? 7. Range 7 ? ? 11. ?(?) is a decreasing function. ? ? = ?? ?? ? 5 Domain is 4 ? 7. Range 4 ? ? 8. ?(?) is a decreasing function. ? ? =?? ? ?? ? ???