Functions

Starter A rule for a function is 𝑓 𝑐 = 1 𝑐+3 Functions KUS objectives BAT understand definition of a function as a one to one mapping BAT recognise odd and even functions BAT state the domain and range of functions Starter A rule for a function is 𝑓 𝑐 = 1 𝑐+3 Evaluate f(2), f(-2), f(0) and f(-5) Which value of f(c) has no solution ? Evaluate f(2), f(-2), f(0) and f(-5)

Notes Domain and Range – mappings Instead of finding a single value of f(x) imagine that each number in the set of possible x values is Input to the function: - the corresponding outputs can be represented as a mapping as shown DOMAIN RANGE Because each element in the first set is mapped to exactly one output we say this mapping is one to one

Notes Domain and Range – mappings Consider this mapping DOMAIN RANGE Because some elements in the first set are mapped to the same output we say this mapping is many to one What other types of mappings can we have? Can you think of any operations that are one to many?

Domain −𝟒<𝐱<𝟓 Range −𝟑<𝒚<𝟑 WB1a Domain and Range – graphically state the domain and range Domain −𝟒<𝐱<𝟓 Range −𝟑<𝒚<𝟑

Sketch a graph to help figure it out Domain: The domain is the set of all possible x-values which will make the function "work", and will output real y-values. Range: The range is the resulting y-values we get after substituting all the possible x-values Sketch a graph to help figure it out Range Domain

Domain −𝟗<𝐱<𝟖 Range 𝟑<𝒚<𝟖 WB1b Domain and Range – graphically state the domain and range Domain −𝟗<𝐱<𝟖 Range 𝟑<𝒚<𝟖

WB2a Sketch the graph of y = f(x) a) f(x) = 𝑥 2 −4𝑥+3, −1<𝑥≤4 What is the domain of f ? What is the range of f ? Range −1<𝑦≤8 Domain −1<𝑥≤4

WB2b Draw a sketch of the function defined by: 𝑓:𝑥⟼ 2𝑥+1, −2<𝑥<4 13−𝑥, 4≤𝑥<10 and state the range of f(x) Range −3<𝑦≤9 Domain −2<𝑥≤10

WB3ab Domain and Range – graphically Sketch each graph and state its domain and range 𝒚= 𝒙 𝟐 −𝟔𝒙+𝟏𝟏 𝐟 𝒙 = 𝟑𝒙+𝟐 𝒙 𝒙=𝟎 𝒚=𝟑 Domain 𝒙∈𝑹 Domain 𝒙∈𝑹 𝒙≠𝟎 Range 𝒚∈𝑹 𝒚>𝟐 Range 𝒚∈𝑹 𝒚≠𝟑

WB3cd Domain and Range – graphically Sketch each graph and state its domain and range g 𝒙 = 𝟏 𝒙 𝟐 𝐡 𝒙 = 𝟏 (𝒙−𝟑)(𝒙+𝟐) 𝒙=𝟎 𝒙=𝟑 𝒙=−𝟐 𝒚=𝟎 Domain 𝒙∈𝑹 𝒙≠𝟎 Domain 𝒙∈𝑹 𝒙≠𝟑 𝒙≠−𝟐 Range 𝒚∈𝑹 𝒚>𝟎 Range 𝒚∈𝑹 𝒚≠𝟎

WB4 The function h(x) is defined by ℎ 𝑥 = 1 𝑥 + 2, 𝑥 𝜖 𝑅 𝑥≠0 a) Sketch a graph of h(x) b) Solve these equations: h(x) = 3 h(x) = 4 h(x) = 1 c) Explain why the equation h(x) = 2 has no solution 𝒙=𝟎 𝒚=𝟐 𝒉 𝒙 = 𝟏 𝒙 +𝟐=𝟑 𝒙=𝟏 𝒉 𝒙 = 𝟏 𝒙 +𝟐=𝟒 𝒙= 𝟏 𝟐 𝒉 𝒙 = 𝟏 𝒙 +𝟐=𝟏 𝒙=−𝟏 𝒉 𝒙 = 𝟏 𝒙 +𝟐=𝟐 𝟏 𝒙 =𝟎 has no solution

WB5 Draw a sketch of the function defined by 𝑔(𝑡)=3𝑡+2, and state it’s domain and range

Notes Problems with one to many

Notes Problems with one to many: geogebra file ‘root (x+a)’ To avoid the problem with one to many – the function 𝑦= 𝑥−1 is usually only defined for positive values

WB6 f(x) = 7−𝑥 , Sketch the graph of y = f(x) What is the domain of f ? What is the range of f ? Range 𝑦≥0 Domain 𝑥≤7

One thing to improve is – KUS objectives BAT understand definition of a function as a one to one mapping BAT recognise odd and even functions BAT state the domain and range of functions self-assess One thing learned is – One thing to improve is –

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