Question

given the function below, fill in the table of values, use the table of values to graph the function, and then identify the functions domain and range.
$y = \log_{4}(x + 3)$
(table of values and graph are shown, then:)
find the domain and range of the function:
domain: blank range: blank

Explanation:

Step1: Analyze the domain of logarithmic function

For a logarithmic function \( y = \log_b(u) \), the argument \( u \) must be greater than 0. In the function \( y=\log_4(x + 3) \), the argument is \( x+3 \). So we set up the inequality \( x + 3>0 \).

Step2: Solve the inequality for x

Subtract 3 from both sides of the inequality \( x + 3>0 \), we get \( x>- 3 \). So the domain is all real numbers greater than - 3, which can be written in interval notation as \( (-3,\infty) \).

Step3: Analyze the range of logarithmic function

The range of a logarithmic function \( y=\log_b(u) \), where \( b>0,b
eq1 \) and \( u>0 \), is all real numbers. This is because as \( u \) takes on all positive real values, \( y=\log_b(u) \) can take on any real number value (it can be made arbitrarily large or arbitrarily small by choosing appropriate values of \( u \)). So for \( y = \log_4(x + 3) \), the range is all real numbers.

Answer:

Domain: \( (-3,\infty) \)
Range: All Real Numbers