If $a(x)=3x+1$ and $b(x)=\sqrt{x-4}$, what is the domain of $(b \circ a)(x)$?
Answer (1)
Find the composite function: ( b ∘ a ) ( x ) = b ( a ( x )) = 3 x − 3 .
Set up the inequality: 3 x − 3 ≥ 0 .
Solve the inequality: x ≥ 1 .
Express the solution in interval notation: [ 1 , ∞ ) .
Explanation
Understanding the Problem We are given two functions, a ( x ) = 3 x + 1 and b ( x ) = x − 4 . We want to find the domain of the composite function ( b ∘ a ) ( x ) . This means we need to find all possible values of x for which the function ( b ∘ a ) ( x ) is defined.
Finding the Composite Function First, let's find the composite function ( b ∘ a ) ( x ) . This is defined as b ( a ( x )) . We substitute a ( x ) into b ( x ) : b ( a ( x )) = b ( 3 x + 1 ) = ( 3 x + 1 ) − 4 = 3 x − 3 .
Setting up the Inequality The domain of a square root function is all values where the expression inside the square root is greater than or equal to 0. Therefore, we need to solve the inequality 3 x − 3 ≥ 0 .
Solving the Inequality To solve the inequality 3 x − 3 ≥ 0 , we first add 3 to both sides: 3 x ≥ 3. Then, we divide both sides by 3: x ≥ 1.
Expressing the Solution in Interval Notation The solution to the inequality is x ≥ 1 . In interval notation, this is [ 1 , ∞ ) .
Final Answer Therefore, the domain of ( b ∘ a ) ( x ) is [ 1 , ∞ ) .
Examples
Imagine you are designing a machine where one component's output ( a ( x ) ) feeds directly into another component ( b ( x ) ). If a ( x ) = 3 x + 1 represents the speed setting of a motor and b ( x ) = x − 4 represents the amount of material processed based on the motor's output, you need to ensure the motor speed is high enough to allow for material processing. Finding the domain of ( b ∘ a ) ( x ) tells you the minimum speed setting ( x ≥ 1 ) for the machine to function correctly, ensuring the material processing component receives a non-negative input.
