Oefening 2.6: Ongedefinieerde en nlereële uitdrukkings 1. Vir watter waarde(s) van $$ m $$ is die volgende uitdrukking a) niereẻel en b) ongedefinieerd? $$ \frac{1}{m^{2}+4 m}-\sqrt{-5-m} $$ 2. Watter beperkings is daar op die volgende vergelyking? $$ \sqrt{p^{2}-1}=p+2 $$ 3. Vir watter waardes van $$ k $$ het die volgende uitdrukking niereële waardes: $$ \frac{1}{k-1}+\sqrt{-k+4 ?} $$ 4. Vir watter $$ x $$-waardes is $$ \frac{x+4}{x^{2}+4} $$ gedefinieer? 5. Vir watter waardes van $$ x $$ is die volgende uitdrukking a) niereëel en b) ongedefinieerd? $$ \sqrt{x^{2}-9}=\frac{1}{x+1} $$

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Let's solve each part of the exercise step by step. ### 1. For which value(s) of $$ m $$ is the expression a) non-real and b) undefined? The expression is given by: $$ \frac{1}{m^{2}+4m}-\sqrt{-5-m} $$ **a) Non-real values:** The square root function is non-real when the expression inside the square root is negative. Therefore, we need to solve: $$ -5 - m < 0 $$ This simplifies to: $$ m > -5 $$ **b) Undefined values:** The expression is undefined when the denominator is zero. Thus, we need to solve: $$ m^{2} + 4m = 0 $$ Factoring gives: $$ m(m + 4) = 0 $$ This results in: $$ m = 0 \quad \text{or} \quad m = -4 $$ ### 2. What restrictions are there on the following equation? The equation is: $$ \sqrt{p^{2}-1} = p + 2 $$ The square root function is defined for non-negative values, so we need: $$ p^{2} - 1 \geq 0 $$ This can be factored as: $$ (p - 1)(p + 1) \geq 0 $$ The critical points are $$ p = -1 $$ and $$ p = 1 $$. Testing intervals gives: - For $$ p < -1 $$: both factors are negative, product is positive. - For $$ -1 < p < 1 $$: one factor is negative, product is negative. - For $$ p > 1 $$: both factors are positive, product is positive. Thus, the solution is: $$ p \leq -1 \quad \text{or} \quad p \geq 1 $$ ### 3. For which values of $$ k $$ does the following expression have non-real values? The expression is: $$ \frac{1}{k-1} + \sqrt{-k + 4} $$ The square root is non-real when: $$ -k + 4 < 0 \implies k > 4 $$ ### 4. For which $$ x $$ values is $$ \frac{x+4}{x^{2}+4} $$ defined? The expression is defined as long as the denominator is not zero: $$ x^{2} + 4 \neq 0 $$ Since $$ x^{2} + 4 $$ is always positive (as $$ x^{2} \geq 0 $$), the expression is defined for all $$ x $$. ### 5. For which values of $$ x $$ is the following expression a) non-real and b) undefined? The expression is: $$ \sqrt{x^{2}-9} = \frac{1}{x+1} $$ **a) Non-real values:** The square root is non-real when: $$ x^{2} - 9 < 0 \implies x^{2} < 9 \implies -3 < x < 3 $$ **b) Undefined values:** The right side is undefined when: $$ x + 1 = 0 \implies x = -1 $$ ### Summary of Results: 1. a) $$ m > -5 $$, b) $$ m = 0 $$ or $$ m = -4 $$ 2. $$ p \leq -1 $$ or $$ p \geq 1 $$ 3. $$ k > 4 $$ 4. Defined for all $$ x $$ 5. a) $$ -3 < x < 3 $$, b) $$ x = -1 $$ **1. a) $$ m > -5 $$** **b) $$ m = 0 $$ or $$ m = -4 $$** **2. $$ p \leq -1 $$ or $$ p \geq 1 $$** **3. $$ k > 4 $$** **4. Defined for all $$ x $$** **5. a) $$ -3 < x < 3 $$** **b) $$ x = -1 $$**

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