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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