Question Statement

Find the domain of the following functions:

(i) $f(x)=x^2-6$

(ii) $g(x)=\frac{x}{x+3}$

(iii) $h(x)=\frac{x+4}{x^2-9}$

(iv) $i(x)=\frac{x}{5x+2}$

(v) $j(x)=\frac{x}{x^2+4}$

(vi) $k(x)=\sqrt{x+1}$

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Background and Explanation

The domain of a function is the set of all possible input values ($x$) for which the function is defined and produces a real number. When finding the domain, we generally look for two main restrictions: denominators cannot be zero, and the expression inside a square root (radicand) must be greater than or equal to zero.

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Solution

Part (i)

Function: $f(x)=x^2-6$

Since $f(x)$ is a polynomial function, it is defined for all real values of $x$. There are no fractions or square roots to restrict the input. Domain: $\mathbb{R}$ (all real numbers).

Part (ii)

Function: $g(x)=\frac{x}{x+3}$

A rational function is undefined when its denominator is zero. We check where $x+3=0$: $g(-3)=\frac{-3}{-3+3}=\frac{-3}{0}$ Since division by zero is undefined ($-\infty$ in the limit context), $x=-3$ must be excluded. Domain: $\mathbb{R}-\{-3\}$

Part (iii)

Function: $h(x)=\frac{x+4}{x^2-9}$

The function $h(x)$ is not defined when the denominator equals zero: $\begin{array}{rl}{x}^2-9& =0\\ x^2& =9\\ x& =\pm 3\end{array}$ The values $3$ and $-3$ make the function undefined. Domain: $\mathbb{R}-\{-3,3\}$

Part (iv)

Function: $i(x)=\frac{x}{5x+2}$

The function $i(x)$ is undefined when the denominator is zero: $\begin{array}{rl}5x+2& =0\\ 5x& =-2\\ x& =\frac{-2}{5}\end{array}$ Domain: $\mathbb{R}-\left\{\frac{-2}{5}\right\}$

Part (v)

Function: $j(x)=\frac{x}{x^2+4}$

We check if the denominator can ever be zero: $x^2+4=0\Rightarrow x^2=-4$. Since the square of any real number is non-negative ($x^2\ge 0$), $x^2+4$ will always be at least $4$. Therefore, $x^2+4\ne 0$ for any real number. Domain: $\mathbb{R}$

Part (vi)

Function: $k(x)=\sqrt{x+1}$

For a square root function to result in a real number, the expression inside the radical must be non-negative: $\begin{array}{rl}x+1& \ge 0\\ x& \ge -1\end{array}$ The function is undefined for any real number less than $-1$. Domain: $[-1,\infty )$

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Key Formulas or Methods Used

• Polynomial Domain: Always $(-\infty ,\infty )$ or $\mathbb{R}$.
• Rational Function Restriction: Set denominator $\ne 0$.
• Square Root Restriction: Set radicand $\ge 0$.
• Set Notation: $\mathbb{R}-\{a\}$ represents all real numbers except $a$.
• Interval Notation: $[a,\infty )$ represents all numbers from $a$ to infinity, including $a$.

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Summary of Steps

1. Identify the function type: Determine if it is a polynomial, rational (fraction), or radical (square root) function.
2. Check for denominators: If there is a denominator, set it equal to zero and solve for $x$. These values are excluded from the domain.
3. Check for square roots: If there is a square root, set the expression inside to $\ge 0$ and solve the inequality.
4. Combine restrictions: State the domain by excluding restricted values from the set of real numbers or using interval notation.