Mathematics

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.

Intuitive explanation

This problem asks you to work with piecewise functions and compose them into a more complex rational expression. The key insight is that $T(x)$ is only defined where its denominator is nonzero—that is, where $f(x)g(x) \neq 1$ —and you'll need to determine which piece of $f$ and which piece of $g$ apply in different regions of the domain. Once you identify the relevant pieces for a given $x$ -value, substitute them into $T(x)$ and simplify, being careful about domain restrictions that make $f$ or $g$ undefined or make the denominator zero.

Intuitive explanation

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