Questions: Which choice is a point of discontinuity on the graph of f(x) = 7/(12x^2 + 2x - 30)? (A) x = -6 (B) x = -10/3 (C) x = 5/3 (D) x = -3/2

Transcript text: Which choice is a point of discontinuity on the graph of $f(x)=\frac{7}{12 x^{2}+2 x-30}$ ? (A) $x=-6$ (B) $x=\frac{-10}{3}$ (C) $x=\frac{5}{3}$ (D) $\times-\frac{3}{2}$

Solution

Find the points of discontinuity of the function $f(x) = \frac{7}{12x^2 + 2x - 30}$.

Factor the denominator.

We have $12x^2 + 2x - 30 = 2(6x^2 + x - 15)$. We look for two numbers that multiply to $(6)(-15) = -90$ and add up to $1$. These numbers are $10$ and $-9$. Thus we can write \[6x^2 + x - 15 = 6x^2 + 10x - 9x - 15 = 2x(3x+5) - 3(3x+5) = (2x-3)(3x+5).\] So, $12x^2 + 2x - 30 = 2(2x-3)(3x+5)$.

Find the zeros of the denominator.

The denominator is zero when $2x-3=0$ or $3x+5=0$. Solving for $x$ we get $x=\frac{3}{2}$ or $x = -\frac{5}{3}$.

Determine the points of discontinuity.

A rational function is discontinuous at the zeros of the denominator. Thus, the points of discontinuity are $x = \frac{3}{2}$ and $x = -\frac{5}{3}$.

$\boxed{\text{The points of discontinuity are } x = \frac{3}{2} \text{ and } x = -\frac{5}{3}.}$

The points of discontinuity are $x = \frac{3}{2}$ and $x = -\frac{5}{3}$. Thus, the answer is (D).

Final Answer: The final answer is $\boxed{D}$

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