Question

use the function ( f(x)=log _{5}(x + 3)-1 ) to answer the following:
(a) determine the equation of the asymptote of ( f(x) ).

(b) determine the domain of ( f(x) ) in interval notation.

(c) determine the range of ( f(x) ) in interval notation.

Explanation:

(a) Determine the equation of the asymptote of \( f(x) \)

Step1: Recall the vertical asymptote of \( y = \log_a(u) \)

The vertical asymptote of \( y=\log_a(u)\) occurs when \(u = 0\). For the function \(y=\log_5(x + 3)-1\), set \(u=x + 3\).

Step2: Solve for \(x\) when \(u = 0\)

If \(x+3=0\), then \(x=-3\).

(b) Determine the domain of \( f(x) \) in interval notation

Step1: Recall the domain of \( y=\log_a(u)\)

The domain of \(y = \log_a(u)\) requires \(u>0\). For \(y=\log_5(x + 3)-1\), we have \(x+3>0\).

Step2: Solve the inequality \(x + 3>0\)

Subtract 3 from both sides: \(x>-3\). In interval notation, this is \((-3,\infty)\).

(c) Determine the range of \( f(x) \) in interval notation

Step1: Recall the range of \( y=\log_a(u)\)

The range of \(y=\log_a(u)\) is \((-\infty,\infty)\). For the function \(y=\log_5(x + 3)-1\), vertical - shift (subtracting 1) does not change the range.

Answer:

(a) \(x=-3\)
(b) \((-3,\infty)\)
(c) \((-\infty,\infty)\)