- Identify the domain of the base function \(f(x) = \log_2 x\) as \((0, \infty)\).
- Recognize that the transformation \(f(x+4)\) shifts the graph horizontally to the left by 4 units.
- Solve the inequality \(x + 4 > 0\) to determine the new domain, resulting in \(x > -4\).
- Conclude that the domain of \(g(x)\) is \((-4, \infty)\), making the statement "domain of \((4, \infty)\)" incorrect.
Final Answer: The statement is False.
Explanation
- Analyze the base function f(x)
The base function is given as \(f(x) = \log_2 x\). The domain of any logarithmic function of the form \(f(x) = \log_b(x)\) is defined by the condition that its argument must be strictly positive, meaning \(x > 0\). Thus, the domain of \(f(x) = \log_2 x\) is \((0, \infty)\).
- Define the transformed function g(x)
The function \(g(x)\) is defined by applying horizontal and vertical transformations to the base function \(f(x)\):
\[g(x) = f(x+4) + 8 = \log_2(x+4) + 8.\]
To determine the domain of \(g(x)\), we must identify the values of \(x\) for which the expression within the logarithm is defined. The argument of the logarithm, \(x+4\), must be greater than zero.
- Determine the domain of g(x)
We set up the inequality for the argument of the logarithm:
\[x + 4 > 0.\]
Subtracting 4 from both sides, we find:
\[x > -4.\]
Therefore, the domain of the function \(g(x)\) consists of all real numbers greater than \(-4\), which is represented in interval notation as \((-4, \infty)\).
- Evaluate the statement
The statement "domain of \((4, \infty)\)" suggests that the function is defined for all \(x > 4\). However, we have determined that the correct domain is \((-4, \infty)\). Thus, the provided statement is incorrect. There are no other features mentioned in the prompt, so the statement "domain of \((4, \infty)\)" is not a correct feature of the function \(g(x)\). The final answer is that this statement is false.
Examples
Logarithmic functions are used in many fields to model phenomena that span multiple orders of magnitude. For example, the pH scale in chemistry is logarithmic, where each whole number change in pH represents a tenfold change in the concentration of hydrogen ions. Understanding how shifting these functions (like changing the domain) affects the model is crucial when adjusting baseline concentrations or recalibrating sensors to measure different ranges of values in scientific experiments.