Question Find the equation(s) of the vertical asymptote(s) of the rational function $g(t) = \frac{t^{2} - 5}{3t^{2} + 4t - 3}$ .

Studdy Solution

STEP 1

1. Vertical asymptotes occur where the denominator of a rational function is equal to zero, provided that the numerator is not also zero at those points.
2. To find the vertical asymptotes, we must factor the denominator and find its roots.
3. The function $g(t)$ is defined for all real numbers except where the denominator is zero.

STEP 2

1. Factor the denominator of $g(t)$ .
2. Find the roots of the denominator.
3. Determine the vertical asymptotes from the roots of the denominator.

STEP 3

Factor the denominator $3t^2 + 4t - 3$ .
$3t^2 + 4t - 3 = (3t - 1)(t + 3)$

STEP 4

Find the roots of the factored denominator by setting each factor equal to zero.
$3t - 1 = 0, \quad t + 3 = 0$

STEP 5

Solve each equation for $t$ to find the roots.
$t = \frac{1}{3}, \quad t = -3$

STEP 6

Determine the vertical asymptotes from the roots of the denominator.
Since the roots of the denominator are $t = \frac{1}{3}$ and $t = -3$ , these are the values where the function $g(t)$ is undefined and where the vertical asymptotes occur.
The equation(s) of the vertical asymptote(s) are:
$t = \frac{1}{3}, \quad t = -3$

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Question Find the equation(s) of the vertical asymptote(s) of the rational function $g(t) = \frac{t^{2} - 5}{3t^{2} + 4t - 3}$ .

Studdy Solution

STEP 1

STEP 2

1. Factor the denominator of $g(t)$ .
2. Find the roots of the denominator.
3. Determine the vertical asymptotes from the roots of the denominator.

STEP 3

Factor the denominator $3t^2 + 4t - 3$ .
$3t^2 + 4t - 3 = (3t - 1)(t + 3)$

STEP 4

Find the roots of the factored denominator by setting each factor equal to zero.
$3t - 1 = 0, \quad t + 3 = 0$

STEP 5

Solve each equation for $t$ to find the roots.
$t = \frac{1}{3}, \quad t = -3$

STEP 6

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Question Find the equation(s) of the vertical asymptote(s) of the rational function g(t)=t2−53t2+4t−3g(t) = \frac{t^{2} - 5}{3t^{2} + 4t - 3}g(t)=3t2+4t−3t2−5​.

Studdy Solution

STEP 1

STEP 2

1. Factor the denominator of g(t)g(t)g(t).2. Find the roots of the denominator.3. Determine the vertical asymptotes from the roots of the denominator.

STEP 3

Factor the denominator 3t2+4t−33t^2 + 4t - 33t2+4t−3.3t2+4t−3=(3t−1)(t+3) 3t^2 + 4t - 3 = (3t - 1)(t + 3) 3t2+4t−3=(3t−1)(t+3)

STEP 4

Find the roots of the factored denominator by setting each factor equal to zero.3t−1=0,t+3=0 3t - 1 = 0, \quad t + 3 = 0 3t−1=0,t+3=0

STEP 5

Solve each equation for ttt to find the roots.t=13,t=−3 t = \frac{1}{3}, \quad t = -3 t=31​,t=−3

STEP 6

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Question Find the equation(s) of the vertical asymptote(s) of the rational function $g(t) = \frac{t^{2} - 5}{3t^{2} + 4t - 3}$ .

1. Factor the denominator of $g(t)$ .
2. Find the roots of the denominator.
3. Determine the vertical asymptotes from the roots of the denominator.

Factor the denominator $3t^2 + 4t - 3$ .
$3t^2 + 4t - 3 = (3t - 1)(t + 3)$

Find the roots of the factored denominator by setting each factor equal to zero.
$3t - 1 = 0, \quad t + 3 = 0$

Solve each equation for $t$ to find the roots.
$t = \frac{1}{3}, \quad t = -3$