select the correct intervals of increase and decrease for the function $f(x)=\\log_4(x + 1)-3$\n\nincrease…

Explanation:

Step1: Determine the domain of the function

For the function (y = \log_{a}u) ((a>0,a\neq1)), the argument (u=x + 1>0), so (x>-1). The domain of (f(x)=\log_{4}(x + 1)-3) is ((-1,\infty)).

Step2: Analyze the monotonicity of the logarithmic function

The general form of a logarithmic function is (y=\log_{a}x). When (a > 1) (in our case (a = 4>1)), the function (y=\log_{a}x) is an increasing function on its domain ((0,\infty)). For the function (y=\log_{4}(x + 1)-3), let (u=x + 1). The function (y=\log_{4}u-3) is a composition of (y=\log_{4}u) and (u=x + 1). The function (u=x + 1) is a linear function with a slope (m = 1>0) (increasing on ((-1,\infty))), and (y=\log_{4}u) is increasing on ((0,\infty)). By the chain - rule for composite functions (if (y = f(g(x))), and (f) and (g) are both increasing or both decreasing, then (y=f(g(x))) is increasing; if one is increasing and the other is decreasing, then (y = f(g(x))) is decreasing), since both (y=\log_{4}u) (with (u=x + 1)) and (u=x + 1) are increasing functions on the domain of (f(x)) ((x>-1)), the function (y=\log_{4}(x + 1)-3) is increasing on its domain ((-1,\infty)).

Answer:

Increase: ((-1,\infty))