What is the domain of $f(x)=4 \sqrt{2 x-1}+3$?
A. $\left(-\infty, \frac{1}{2}\right]$
B. $\left(-\infty,-\frac{1}{2}\right]$
C. $\left[-\frac{1}{2}, \infty\right)$
D. $\left[\frac{1}{2}, \infty\right)$
Answer (1)
The domain of f ( x ) = 4 2 x − 1 + 3 is found by ensuring the expression inside the square root is non-negative.
Set up the inequality 2 x − 1 ≥ 0 .
Solve for x to get x ≥ 2 1 .
Express the solution in interval notation: [ 2 1 , ∞ ) .
Explanation
Understanding the Problem We are given the function f ( x ) = 4 2 x − 1 + 3 and asked to find its domain. The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this case, we have a square root, and the expression inside the square root must be non-negative to produce a real number result.
Setting up the Inequality To find the domain, we need to determine the values of x for which the expression inside the square root is greater than or equal to zero. This gives us the inequality: 2 x − 1 ≥ 0
Solving the Inequality Now, we solve the inequality for x :
Add 1 to both sides: 2 x ≥ 1 Divide both sides by 2: x ≥ 2 1
Expressing the Solution in Interval Notation The solution to the inequality is x ≥ 2 1 . This means that the domain of the function f ( x ) is all real numbers x such that x is greater than or equal to 2 1 . In interval notation, this is written as [ 2 1 , ∞ ) .
Final Answer Therefore, the domain of the function f ( x ) = 4 2 x − 1 + 3 is [ 2 1 , ∞ ) .
Examples
Consider a scenario where you are designing a garden and need to determine the minimum amount of fertilizer required for your plants to thrive. If the growth rate of a plant is modeled by the function f ( x ) = 4 2 x − 1 + 3 , where x represents the amount of fertilizer used, you need to ensure that x is within the domain of the function to get a meaningful result. In this case, you need at least 2 1 unit of fertilizer for the plant to start growing according to the model. Understanding the domain helps you avoid using amounts of fertilizer that would lead to invalid or meaningless predictions about plant growth.
