Question

Find the domain of \(f(x)=2 \sqrt{3-5 x}-4\)

Short answer

The domain is \((-\infty, \frac{3}{5}]\).

Step-by-step solution

Understand the function

The given function is \(f(x) = 2 \sqrt{3 - 5x} - 4\). To find the domain, determine the values of \(x\) for which the function is defined.

Identify restrictions from the square root

The expression inside the square root \(3 - 5x\) must be non-negative because the square root of a negative number is not defined. Therefore, set up the inequality: \(3 - 5x \geq 0\).

Solve the inequality

Solve the inequality \(3 - 5x \geq 0\): 1. Subtract 3 from both sides: \( -5x \geq -3 \).2. Divide by -5 (and reverse the inequality): \( x \leq \frac{3}{5} \).

Write the domain in interval notation

The domain of \(f(x)\) is all \(x\) values that satisfy the inequality. In interval notation: \( (-\infty, \frac{3}{5}] \).

Key concepts

inequalities

Inequalities help us understand which values of a variable make an expression true. For example, in the function given by the exercise, we need to determine when the expression inside the square root is non-negative. We use the inequality grade school mathematics has already taught you: solving linear inequalities involves both subtraction and division operations. Just remember, when you divide or multiply by a negative number, the direction of the inequality sign flips.

square roots

Square roots are mathematical operations that find a number which, when multiplied by itself, gives the initial value. In our function, we have a square root of an expression, specifically \( \sqrt{3 - 5x}\).
Since the square root of a negative number is not defined in the set of real numbers, we need to ensure that \(3 - 5x \geq 0\). This requirement leads us directly to the inequalities we solved previously. For any function containing a square root, determining which parts of the expression must be non-negative is crucial to finding its domain.

interval notation

Interval notation is a way of writing subsets of the real number line using intervals. When we express domains, we use square brackets \[\]\ for inclusive boundaries and parentheses \(\)\ for exclusive boundaries. Given the inequality \(x \leq \frac{3}{5}\), we translate it into interval notation as follows: \(-\infty, \frac{3}{5}]\).
This shows all real numbers from negative infinity up to and including \(\frac{3}{5}\). Learning how to convert inequalities into interval notation helps in clearly and concisely expressing the domain of functions.