To find the domain of the function [tex]\( f(x) = \sqrt{-3x + 12} \)[/tex], we need to determine the values of [tex]\( x \)[/tex] for which the expression inside the square root is non-negative. The square root function is defined only when its argument is greater than or equal to zero.
So, we start by setting up the inequality:
[tex]\[ -3x + 12 \geq 0 \][/tex]
To solve this inequality, we need to isolate [tex]\( x \)[/tex]. Follow these steps:
1. Subtract 12 from both sides:
[tex]\[ -3x + 12 - 12 \geq 0 - 12 \][/tex]
[tex]\[ -3x \geq -12 \][/tex]
2. Divide both sides by -3. When we divide or multiply an inequality by a negative number, the direction of the inequality sign reverses:
[tex]\[ \frac{-3x}{-3} \leq \frac{-12}{-3} \][/tex]
[tex]\[ x \leq 4 \][/tex]
Thus, the values of [tex]\( x \)[/tex] that satisfy the inequality are all [tex]\( x \)[/tex] such that [tex]\( x \leq 4 \)[/tex].
In interval notation, we write this as:
[tex]\[ (-\infty, 4] \][/tex]
So the domain of the function [tex]\( f(x) = \sqrt{-3x + 12} \)[/tex] is:
[tex]\[ (-\infty, 4] \][/tex]
