Questions: Given the graph of the Logarithmic Function g(x)=log2 x : The graph of the function f(x)=7+log2(x) can be obtained from the graph of g(x) by shifting the graph of g(x) 7 units; Identify the Domain and Range of f(x) using Interval Notation. Domain of the function f(x) is Range of the function f(x) is

Given the graph of the Logarithmic Function g(x)=log2 x :
The graph of the function f(x)=7+log2(x) can be obtained from the graph of g(x) by shifting the graph of g(x) 7 units;

Identify the Domain and Range of f(x) using Interval Notation.
Domain of the function f(x) is
Range of the function f(x) is

Transcript text: Given the graph of the Logarithmic Function $g(x)=\log _{2} x$ : The graph of the function $f(x)=7+\log _{2}(x)$ can be obtained from the graph of $g(x)$ by shifting the graph of $g(x)$ Select an answer $\boldsymbol{0} 7$ units; Identify the Domain and Range of $f(x)$ using Interval Notation. Domain of the function $f(x)$ is $\square$ Range of the function $f(x)$ is $\square$

Solution

Solution Steps

Solution Approach

To determine the domain and range of the function $ f(x) = 7 + \log_2(x) $, we start by considering the domain and range of the base function $ g(x) = \log_2(x) $. The domain of $ g(x) $ is $ (0, \infty) $ because logarithms are only defined for positive numbers. The range of $ g(x) $ is $ (-\infty, \infty) $ because a logarithmic function can take any real value. Since $ f(x) $ is a vertical shift of $ g(x) $ by 7 units, the domain remains the same, $ (0, \infty) $, and the range is also $ (-\infty, \infty) $.

Step 1: Determine the Domain of $ f(x) $

The function $ f(x) = 7 + \log_2(x) $ is defined for values of $ x $ where the logarithm is valid. Since the logarithmic function $ \log_2(x) $ is only defined for $ x > 0 $, the domain of $ f(x) $ is given by: \[ \text{Domain of } f(x): (0, \infty) \]

Step 2: Determine the Range of $ f(x) $

The range of the function $ g(x) = \log_2(x) $ is $ (-\infty, \infty) $. Since $ f(x) $ is obtained by shifting $ g(x) $ vertically by 7 units, the range of $ f(x) $ remains the same as that of $ g(x) $: \[ \text{Range of } f(x): (-\infty, \infty) \]