Find the domain of the function. [tex]f(x)=\sqrt{3 x-12}[/tex] Write your answer in interval notation. Use inf for [tex]$\infty$[/tex].
Answer (2)
Set the expression inside the square root greater than or equal to zero: 3 x − 12 g e 0 .
Solve the inequality for x : x g e 4 .
Express the solution in interval notation: [ 4 , in f t y ) .
The domain of the function is [ 4 , in f ] .
Explanation
Understanding the Problem We are given the function f ( x ) = 3 x − 12 and we want 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. Since we have a square root, the expression inside the square root must be greater than or equal to zero.
Solving the Inequality To find the domain, we need to solve the inequality 3 x − 12 g e 0 . Let's add 12 to both sides of the inequality: 3 x − 12 + 12 g e 0 + 12 3 x g e 12 Now, divide both sides by 3: 3 3 x g e 3 12 x g e 4
Expressing the Solution in Interval Notation The solution to the inequality is x g e 4 . This means that the function is defined for all x values greater than or equal to 4. In interval notation, this is written as [ 4 , in f t y ) . We use a square bracket on the left because 4 is included in the domain, and we use 'inf' to represent infinity.
Final Answer Therefore, the domain of the function f ( x ) = s q r t 3 x − 12 is [ 4 , in f t y ) .
Examples
Understanding the domain of a function is crucial in many real-world applications. For example, if f ( x ) represents the number of items sold as a function of the price x , then knowing the domain tells us the range of prices for which the model is valid. If the domain is [ 4 , ∞ ) , it means the price must be at least 4 f or t h e m o d e lt o mak ese n se . S imi l a r l y , in p h ys i cs , i f f(x) re p rese n t s t h e d i s t an ce an o bj ec tt r a v e l s in t im e x , t h e d o main w o u l d t e ll u s t h e v a l i d t im e in t er v a l s f or w hi c h t h e d i s t an ce f u n c t i o ni s a ppl i c ab l e . I n f inan ce , i f f(x) re p rese n t s t h e p ro f i t a s a f u n c t i o n o f in v es t m e n t x$, the domain specifies the investment amounts for which the profit function is meaningful.
The domain of the function f ( x ) = 3 x − 12 is [ 4 , ∞ ) . This indicates the function is defined for all values of x greater than or equal to 4. Thus, any substitution of x in this range will yield a valid output for the function.
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