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Consider the function [tex]\( f(t) = \sqrt{3t - 9} \)[/tex].

To ensure the expression inside the square root is non-negative, solve the inequality:
[tex]\[ 3t - 9 \geq 0 \][/tex]

Solve for [tex]\( t \)[/tex].


Sagot :

To determine the domain of the function [tex]\( f(t) = \sqrt{3t - 9} \)[/tex], we need to make sure that the expression inside the square root, [tex]\( 3t - 9 \)[/tex], is non-negative (i.e., greater than or equal to zero). This is because the square root function is only defined for non-negative numbers.

Here are the steps to find the domain:

1. Define the inequality:
[tex]\[ 3t - 9 \geq 0 \][/tex]

2. Solve the inequality for [tex]\( t \)[/tex]:
- Add 9 to both sides to isolate the term with [tex]\( t \)[/tex]:
[tex]\[ 3t \geq 9 \][/tex]
- Divide both sides by 3 to solve for [tex]\( t \)[/tex]:
[tex]\[ t \geq \frac{9}{3} \][/tex]
[tex]\[ t \geq 3 \][/tex]

So, for the function [tex]\( f(t) = \sqrt{3t - 9} \)[/tex] to be defined, [tex]\( t \)[/tex] must be greater than or equal to 3.

Therefore, the domain of the function is:
[tex]\[ t \geq 3 \][/tex]
In interval notation, the domain can be expressed as:
[tex]\[ [3, \infty) \][/tex]

Thus, the minimum value of [tex]\( t \)[/tex] for which the function [tex]\( f(t) = \sqrt{3t - 9} \)[/tex] is defined is [tex]\( 3 \)[/tex].