Pure Mathematics · 2nd secondary · First Term

Functions of a Real Variable and Drawing Curves — The real functions

About this lesson

This unit introduces the concept of real functions, focusing on identifying functions from graphs using the vertical line test, and determining the domain and range of functions from their graphs or algebraic rules. It covers piecewise-defined functions, explaining how to graph them and deduce their domain and range. The unit also addresses operations on functions (addition, subtraction, multiplication, and division), including how to find the domain of the resulting function, and introduces composition of functions with examples and exercises. Students are expected to be able to graph functions, determine domains and ranges, perform operations on functions, and evaluate composite functions. The text includes worked examples, 'Try to solve' exercises, and application problems in contexts such as mechanics and trade.

Main topics in this lesson

  • Real functions and the vertical line test
  • Domain and range of real functions
  • Piecewise-defined functions and their graphs
  • Operations on functions (addition, subtraction, multiplication, division)
  • Composition of functions

Key terms and vocabulary

  • real function
  • domain
  • co-domain
  • range
  • vertical line test
  • arrow diagram
  • Cartesian diagram
  • piecewise function
  • operations on functions
  • composition of functions

Questions and answers on Functions of a Real Variable and Drawing Curves — The real functions - practice and revision

Multiple choice questions on this lesson in Pure Mathematics for 2nd secondary, First Term, with practice exercises for revision and exam preparation.

1

الدالة د(س) = س² + ١ مجالها ح، فإن مداها هو:

  • 1 [١، ∞)
  • 2 (١، ∞)
  • 3 (−∞، ١]
  • 4 ح
2

إذا كانت د(س) = ٢س + ٣، فإن د(د(١)) تساوي:

  • 1 ١٣
  • 2 ٥
  • 3 ٨
  • 4 ١٠
3

إذا كانت د(س) = س + ١ و ق(س) = س − ١، فإن (د/ق)(س) تساوي:

  • 1 (س + ١)/(س − ١) ومجالها ح − {١}
  • 2 (س + ١)/(س − ١) ومجالها ح
  • 3 (س − ١)/(س + ١) ومجالها ح − {−١}
  • 4 (س + ١)/(س − ١) ومجالها ح − {−١}
4

إذا كانت د(س) = ٤س − ١ و ق(س) = س²، فإن (د − ق)(٢) تساوي:

  • 1 ٣
  • 2 ٧
  • 3 ١١
  • 4 −٣
5

أي من العلاقات الآتية لا تمثل دالة حقيقية؟

  • 1 س = ص²
  • 2 ص = س²
  • 3 ص = ٥
  • 4 ص = −٣س + ١
6

إذا كانت د(س) = س² − ١، فإن مدى الدالة عندما يكون المجال هو {−٢، ٠، ٢} هو:

  • 1 {٣، −١}
  • 2 {٣، ٠، −١}
  • 3 {٤، ٠}
  • 4 {−١، ٣، ٤}

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