Question $\newline$ Find all vertical asymptotes of the following function. $\newline$ $f(x)=\frac{x-4}{3 x-3}$ $\newline$ Answer Attempt $1$ out of $3$

Full solution

Q. Question

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Find all vertical asymptotes of the following function.

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f(x)=\frac{x-4}{3 x-3}

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Answer Attempt

1

out of

3

Identify potential vertical asymptotes: Identify the potential vertical asymptotes. Vertical asymptotes occur where the denominator of a rational function is equal to $0$ , as long as the numerator is not also $0$ at those points. For the function $f(x) = \frac{x-4}{3x-3}$ , we need to find the values of $x$ that make the denominator $3x-3$ equal to $0$ .
Solve for x-values: Solve the equation $3x-3 = 0$ to find the x-values. $\newline$ To find the x-values that make the denominator zero, we solve the equation: $\newline$ $3x - 3 = 0$ $\newline$ Adding $3$ to both sides gives: $\newline$ $3x = 3$ $\newline$ Dividing both sides by $3$ gives: $\newline$ $x = 1$
Check numerator at $x=1$ : Check if the numerator is also zero at $x = 1$ . $\newline$ We need to ensure that the numerator, $x-4$ , is not also zero when $x = 1$ . Plugging $x = 1$ into the numerator gives: $\newline$ $1 - 4 = -3$ $\newline$ Since $-3$ is not equal to zero, the numerator is not zero when $x = 1$ .
Conclude vertical asymptote location: Conclude the location of the vertical asymptote. Since the denominator is $0$ and the numerator is not $0$ at $x = 1$ , there is a vertical asymptote at $x = 1$ .