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Basic Functions and Relations

This tutorial covers material encountered in chapter 1 of the VCE Mathematical Methods Textbook, namely:

  • The domain and range of basic relations/functions
  • The maximal domain of a function
  • The sum and product of functions
  • Compositions of functions
  • Inverses of functions
  • Basic power functions

Q1 – Domain and Range of Function and Relation

Q2 – Domain and Range of Function and Inverse

Q3 – Maximal Domains of Functions

Q4, 5 & 6 – Sum, Product, Inverses and Compositions of Functions

Worksheet

Q1. For each of the following relations state the implied domain and range:

(a) f(x)=x^2 + 3

(b) f(x)=3x-2

(c) \{(x,y):x^2+y^2=9\}

(d) \{(x,y):y\geq2x+1\}

Q2. For the function g:[0,5] \to \R ,\,g(x)=\dfrac{x-4}{5}

(a) State the range of g.

(b) Find g^{-1}, and state the domain and range of g^{-1}.

(c) Find \{x:g(x)=2\}

(d) Find \{x:g^{-1}(x)=4\}

Q3. Find the implied domain for each of the following:

(a) f(x)=\dfrac{1}{3x-1}

(b) g(x)=\dfrac{1}{\sqrt{x^2-9}}

(c) h(x)=\dfrac{1}{(x+3)(x-2)}

(d) j(x)=\sqrt{9-x^2}

Q4. For f(x)=(x-2)^2 and g(x) = x + 4, find (f + g)(x) and (fg)(x)

Q5. Find the inverse of each of the following functions:

(a) f: \R \to \R,\,f(x)=x^3

(b) f: (-\infty,0]\to\R,\,f(x)=2x^5

(c) f:(1,\infty)\to\R,\,f(x)=10000x^4

Q6. For f(x) = 3x + 1 and g(x) = x^3 + 1, find:

(a) f\circ g(x)

(b) g\circ f(x)

(c) g\circ g(x)

(d) f \circ f(x)

(e) f \circ (f+g)(x)

(f) f \circ (fg)(x)

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