Let [tex]f(x) = \sqrt{x-3}[/tex] and [tex]g(x) = 9[/tex]. Find the domain of the composite function [tex](g \circ f)(x)[/tex].
A company uses a pricing function given by [tex]f(x) = 2x+5[/tex], where x is the number of items purchased and [tex]f(x)[/tex] is the total cost in dollars.
(a) Find the inverse function [tex]f^{-1}(x)[/tex].
(b) Interpret the meaning of [tex]f^{-1}(25)[/tex] in this context.
Answer (2)
The domain of the composite function ( g ∘ f ) ( x ) is [ 3 , ∞ ) . The inverse function of f ( x ) = 2 x + 5 is f − 1 ( x ) = 2 x − 5 . The interpretation of f − 1 ( 25 ) = 10 implies that 10 items were purchased for a total cost of 25 dollars.
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The domain of the composite function ( g ∘ f ) ( x ) is found by ensuring the expressions inside the square roots are non-negative, resulting in x ≥ 12 .
The inverse function f − 1 ( x ) of f ( x ) = 2 x + 5 is found by swapping x and y and solving for y , giving f − 1 ( x ) = 2 x − 5 .
The value f − 1 ( 25 ) is calculated as 2 25 − 5 = 10 .
f − 1 ( 25 ) = 10 means that 10 items were purchased for a total cost of $25, demonstrating the application of the inverse function in a real-world context.
Explanation
Understanding the Problem We are given two functions: f ( x ) = x − 3 and g ( x ) = 9 f ( x ) . We need to find the domain of the composite function ( g ∘ f ) ( x ) = g ( f ( x )) . Additionally, we are given a pricing function f ( x ) = 2 x + 5 , where x is the number of items purchased and f ( x ) is the total cost in dollars. We need to find the inverse function f − 1 ( x ) and interpret the meaning of f − 1 ( 25 ) in this context.
Finding the Composite Function First, let's find the composite function g ( f ( x )) . Since g ( x ) = 9 f ( x ) , we have g ( f ( x )) = 9 f ( x ) − 3 = 9 x − 3 − 3 .
Determining the Domain Now, let's determine the domain of g ( f ( x )) . For the outer square root to be defined, we need x − 3 − 3 ≥ 0 . This implies x − 3 ≥ 3 . Squaring both sides, we get x − 3 ≥ 9 , so x ≥ 12 . Also, for the inner square root to be defined, we need x − 3 ≥ 0 , so x ≥ 3 . Since x ≥ 12 satisfies x ≥ 3 , the domain of ( g ∘ f ) ( x ) is x ≥ 12 .
Finding the Inverse Function Next, let's find the inverse function f − 1 ( x ) for f ( x ) = 2 x + 5 . Let y = 2 x + 5 . To find the inverse, we solve for x in terms of y : y = 2 x + 5 ⇒ 2 x = y − 5 ⇒ x = 2 y − 5 . Therefore, f − 1 ( x ) = 2 x − 5 .
Interpreting the Inverse Function Finally, let's interpret the meaning of f − 1 ( 25 ) . We have f − 1 ( 25 ) = 2 25 − 5 = 2 20 = 10 . In the context of the problem, f − 1 ( 25 ) represents the number of items purchased when the total cost is 25. S o , f^{-1}(25) = 10$ means that 10 items were purchased for a total cost of $25.
Examples
Imagine you're baking cookies for a bake sale. The composite function helps determine the minimum amount of ingredients you need to start with to ensure you have enough after a series of preparations. The inverse function, on the other hand, is useful for budgeting. If you know how much money you have and the pricing function, the inverse function tells you exactly how many cookies you can afford to buy. This kind of problem-solving is crucial in managing resources and making informed decisions in everyday scenarios.
