If $g(x)=\frac{x+1}{x-2}$ and $h(x)=4-x$, what is the value of $(g \circ h)(-3)$?
Answer (1)
First, find h ( − 3 ) by substituting − 3 into h ( x ) : h ( − 3 ) = 4 − ( − 3 ) = 7 .
Then, find g ( h ( − 3 )) by substituting 7 into g ( x ) : g ( 7 ) = 7 − 2 7 + 1 = 5 8 .
Thus, ( g ∘ h ) ( − 3 ) = 5 8 .
The value of ( g ∘ h ) ( − 3 ) is 5 8 .
Explanation
Understanding the Problem We are given two functions, g ( x ) = x − 2 x + 1 and h ( x ) = 4 − x . Our goal is to find the value of the composite function ( g ∘ h ) ( − 3 ) , which means we need to find g ( h ( − 3 )) .
Evaluating h(-3) First, we need to evaluate h ( − 3 ) . Substituting x = − 3 into the expression for h ( x ) , we get: h ( − 3 ) = 4 − ( − 3 ) = 4 + 3 = 7
Evaluating g(h(-3)) Now that we have h ( − 3 ) = 7 , we can substitute this value into the function g ( x ) to find g ( h ( − 3 )) = g ( 7 ) .
g ( 7 ) = 7 − 2 7 + 1 = 5 8
Final Answer Therefore, ( g ∘ h ) ( − 3 ) = g ( h ( − 3 )) = 5 8 .
Examples
Composite functions are used in many real-world applications. For example, in manufacturing, if h ( x ) represents the number of raw materials needed to produce x units of a product, and g ( x ) represents the cost of x units of raw materials, then ( g ∘ h ) ( x ) represents the cost of raw materials needed to produce x units of the final product. Understanding composite functions helps businesses optimize their production processes and manage costs effectively.
