Question

State the domain, vertical asymptote, and end behavior of the function f(x) = log5 (39 − 13x) + 7.

Answer 1

Consider the following logarithmic function;

f(x) = log5(39 – 13x) + 7

 

The domain and the vertical asymptote of the function are obtained as follows:

The domain of the logarithmic function is;

The logarithmic function is defined only when the input is positive,

So, the function is defined as;

        39 – 13x > 0

39 – 39 – 13x > - 39

                     x > -39/-13

                     x > 3

 

Hence the domain of the function is (3, ∞).

 

The vertical asymptote of the function is as follow;

        39 – 13x = 0

39 – 39 – 13x = -39

                     x = -39/-13

                     x = 3

 

Therefore, the vertical asymptote of the function is x = 3.

 

The end of the behavior for x:(-∞ → ∞) is explained as follows:

limx→-∞f(x) = 39 – 13x

                      = 3

 

Similarly;

limx→∞f(x) = 39 – 13x

                     =  ∞

 

Therefore,

 x → (3)-, f(x) → -∞ and x → ∞, f(x) = ∞.

Therefore,

 x → (3)-, f(x) → -∞ and x → ∞, f(x) = ∞.