Question

Multiple Choice Which of the following is the domain of \(f^{\prime}(x)\) if \(f(x)=\log _{2}(x+3) ? \quad\) (A) \(x<-3 \quad\) (B) \(x \leq 3 \quad\) (C) \(x \neq-3 \quad\) (D) \(x>-3\) (E) \(x \geq-3\)

Short answer

The domain of \(f^{\prime}(x)\) if \(f(x)=\log _{2}(x+3)\) is \(x>-3\), corresponding to option (D).

Step-by-step solution

Identify the function's domain

The function \(f(x)=\log _{2}(x+3)\) is logarithmic and its domain are the \(x\) values that make the argument \(x+3\) positive (since you can't take the logarithm of a negative number or zero). This results in \(x>-3\). Essentially, the function is defined for all real numbers greater than -3.

Match the obtained domain with the options

The obtained domain is \(x>-3\). Looking at the choices, it corresponds to the option (D) \(x>-3\).

Key concepts

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions and are of fundamental importance in various fields, including mathematics, science, and engineering. A logarithmic function can be written generally as \( f(x) = \log_b(x) \), where \( b \) is the base of the logarithm and \( x \) is the argument. The base \( b \) is typically a positive real number.

The function takes as input a positive number \( x \) and gives the power to which the base \( b \) must be raised to obtain \( x \). It's crucial to note that the logarithm of a non-positive number is undefined, which means that the argument \( x \) must always be greater than zero for the logarithmic function to be properly defined.

• For example, in the function \( \log_2(x+3) \), the base is 2 and the argument is \( x+3 \).
• Here, we need \( x+3 > 0 \), which simplifies to \( x > -3 \) in order to keep the argument of the logarithm positive.

This restriction on the value of \( x \) is essential when determining the domain of logarithmic functions.

Finding Domain

The domain of a function refers to the set of all possible input values (often represented as \( x \) values) for which the function is defined. To find the domain of a logarithmic function, you need to ensure that the argument of the logarithm is strictly greater than zero.

In the case of \( f(x) = \log_2(x+3) \), the domain would be the solution to the inequality \( x+3 > 0 \).

Steps for Finding the Domain

• Identify the Argument: Look inside the logarithm to find the argument.
• Solve the Inequality: Make sure the argument is positive by solving the inequality.
• Write the Domain: Express the domain in interval notation or inequality form.

The domain answers the question of which \( x \)-values can be safely plugged into the function without causing mathematical errors such as taking the logarithm of a non-positive number.

Derivatives of Functions

The derivative of a function at a point provides the rate at which the function value changes with respect to its input value. In other words, it measures the function's sensitivity to change in its input (often time or position). Derivatives are a central concept in calculus and are used to find slopes, velocities, and rates of change.

To find the derivative of a logarithmic function, like \( f(x) = \log_b(x) \), where \( b \) is the base of the logarithm, apply the logarithmic differentiation rules. If \( f(x) = \log_b(x+3) \), we consider the domain of \( f(x) \) to determine the domain of its derivative, denoted as \( f'(x) \).

Important Points About Derivatives

• The derivative exists only where the function is differentiable, which generally requires that the function is continuous.
• The domain of the derivative may differ from the domain of the function itself, particularly in cases where the function's domain boundary corresponds to a sharp corner or cusp in its graph.
• For logarithmic functions, the domain of the derivative will typically be the same as the domain of the function itself, as these functions don't normally have corners or discontinuities within their domains.

For the given function \( f(x) = \log_2(x+3) \), we first establish its domain before finding its derivative to ensure we're considering values where the derivative exists.