classify the functions as continuous or discontinuous. f(x)=(x^2 - 14x + 24)/(x^2 + 3x - 28) g(x)=(x^2 - 2x…

Explanation:

Step1: Recall continuity condition

A rational - function $y = \frac{p(x)}{q(x)}$ is discontinuous at $x = a$ if $q(a)=0$.

Step2: Analyze $f(x)=\frac{x^{2}-14x + 24}{x^{2}+3x - 28}$

Factor $x^{2}+3x - 28=(x + 7)(x - 4)$. When $x=-7$ or $x = 4$, $q(x)=0$, so $f(x)$ is discontinuous.

Step3: Analyze $g(x)=\frac{x^{2}-2x - 48}{x^{2}-64}$

Factor $x^{2}-64=(x + 8)(x - 8)$. When $x=-8$ or $x = 8$, $q(x)=0$, so $g(x)$ is discontinuous.

Step4: Analyze $h(x)=\frac{x + 7}{x^{2}-x + 4}$

The discriminant of $x^{2}-x + 4$ is $\Delta=(-1)^{2}-4\times4=1 - 16=-15<0$. So $x^{2}-x + 4\neq0$ for all real $x$, and $h(x)$ is continuous.

Step5: Analyze $i(x)=\frac{x^{2}+21x + 104}{x + 13}$

Factor $x^{2}+21x + 104=(x + 8)(x+13)$. When $x=-13$, $q(x)=0$, so $i(x)$ is discontinuous.

Step6: Analyze $j(x)=\frac{x^{2}-16}{x^{2}+16}$

The denominator $x^{2}+16>0$ for all real $x$ since $x^{2}\geq0$ for all real $x$. So $j(x)$ is continuous.

Step7: Analyze $k(x)=\frac{2x - 10}{x^{2}+9}$

The denominator $x^{2}+9>0$ for all real $x$ since $x^{2}\geq0$ for all real $x$. So $k(x)$ is continuous.

Step8: Analyze $l(x)=\frac{x^{2}+x}{x^{2}+x + 25}$

The discriminant of $x^{2}+x + 25$ is $\Delta=1^{2}-4\times25=1 - 100=-99<0$. So $x^{2}+x + 25\neq0$ for all real $x$, and $l(x)$ is continuous.

Step9: Analyze $m(x)=\frac{x^{2}-196}{x^{2}+14}$

The denominator $x^{2}+14>0$ for all real $x$ since $x^{2}\geq0$ for all real $x$. So $m(x)$ is continuous.

Answer:

Continuous	Discontinuous
$h(x),j(x),k(x),l(x),m(x)$	$f(x),g(x),i(x)$