Find the domain and vertical asymptote of f ( x ) = − log ⁡ 2 x + 3 f ( x ) = - \log _ { 2 } x + 3 f ( x ) = − lo g 2 x + 3 .A) Domain: ( 0 , ∞ ) ( 0 , \infty ) ( 0 , ∞ ) Asymptote: y -axisB) Domain: ( 0 , ∞ ) ( 0 , \infty ) ( 0 , ∞ ) Asymptote: x=3C) Domain: ( 3 , ∞ ) ( 3 , \infty ) ( 3 , ∞ ) Asymptote: x = 8.00 x = 8.00 x = 8.00 D) Domain: ( − ∞ , ∞ ) ( - \infty , \infty ) ( − ∞ , ∞ ) Asymptote: y -axisE) Domain: ( 3 , ∞ ) ( 3 , \infty ) ( 3 , ∞ ) Asymptote: y -axis

Question

Find the domain and vertical asymptote of   f ( x )  = − log ⁡ 2 x + 3 f ( x )  = - \log _ { 2 } x + 3 f ( x )  = − lo g 2 ​ x + 3   .A) Domain:   ( 0 , ∞ )  ( 0 , \infty )  ( 0 , ∞ )    Asymptote: y -axisB) Domain:   ( 0 , ∞ )  ( 0 , \infty )  ( 0 , ∞ )    Asymptote: x=3C) Domain:   ( 3 , ∞ )  ( 3 , \infty )  ( 3 , ∞ )    Asymptote:   x = 8.00 x = 8.00 x = 8.00 D) Domain:   ( − ∞ , ∞ )  ( - \infty , \infty )  ( − ∞ , ∞ )    Asymptote: y -axisE) Domain:   ( 3 , ∞ )  ( 3 , \infty )  ( 3 , ∞ )    Asymptote: y -axis

Charisse Blackwell · Accepted Answer

Final Answer : A, Explanation :  The function is defined for all positive values of $x$ since the logarithm is not defined for $x \leq 0$. The vertical asymptote occurs when the argument of the logarithm approaches $0$, which means $x$ approaches $0$. This leads to the function becoming infinitely negative, but since there is a positive constant of $3$ added to the function, the asymptote will be the $y$-axis. Therefore, the answer is A.

Find the domain and vertical asymptote of $f ( x ) = - \log _ { 2 } x + 3$ .

A) Domain: $\infty )$ Asymptote: y -axis
B) Domain: $\infty )$ Asymptote: x=3
C) Domain: $\infty )$ Asymptote: $x = 8.00$
D) Domain: $\infty , \infty )$ Asymptote: y -axis
E) Domain: $\infty )$ Asymptote: y -axis

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