Basic Functions and Relations [Video Tutorial]

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

Got questions? Share and ask here...