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Determine the domain, range, and vertical asymptote of function below. show all work.f(x)=log5(x+4) |

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Answer:

You need to remember the condition for the logarithm of a number such that the number is strictly positive, hence `x+4 gt 0` .

You need to solve the inequality `x + 4 gt 0`  to find the domain of the function.

`x + 4 gt 0 =gt x gt -4`  => `x in (-4,oo)`

`` Hence, the interval `(-4,oo)`  denotes the domain of the function `f(x) = log_5 (x+4)` .

You need to find the range of the function, hence you need to remember that the logarithmic function is the inverse of exponential function and the domain of exponential function is the range of logarithmic function. Since the domain of exponential function comprises all real values, hence the range of the function `f(x) = log_5 (x+4)`  consists of all real values.

You need to solve the equation x + 4 = 0 to get the vertical asymptote.

`x + 4 = 0 =gt x = -4`  => the vertical asymptote of the function `f(x) = log_5 (x+4)`  is x = -4.

Hence, the domain of the function is `(-4,oo), ` the range is real set R and the vertical asymptote is `x = -4` .

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