The functions and are defined as follows: f(x)= cases 1 x-1 , & x3

MATHEMATICS · Multi-part · A Level · Mathematics · Algebra and Functions · Algebra and Functions

Problem

The functions

f

and

g

are defined as follows:

f(x)=\begin{cases} \dfrac{1}{x-1}, & x<2,\\ \sqrt{6-x}, & x\ge 2, \end{cases} \quad g(x)=\begin{cases} \log_{e}(x-3), & x>3,\\ x^{2}, & x\le 3. \end{cases}

Define the function

T(x)=\dfrac{\bigl(f(x)-g(x)\bigr)\,g(x)}{f(x)\,g(x)-1}.

Answer the following. (a) Part 1: (i) State the domain of

f

. Give your answer using set notation. (b) Part 2: (ii) State the domain of

g

. (c) Part 3: (iii) Determine the set of real numbers for which the numerator

(f(x)-g(x))\,g(x)

is defined. (d) Part 4: (iv) Determine the set of real numbers for which the denominator

f(x)\,g(x)-1

is defined and non-zero. (e) Part 5: (v) Hence, obtain the complete domain of

T

. Present your answer in set-builder form.