If $g(x)=\frac{x+1}{x-2}$ and $h(x)=4-x$, what is the value of $(g \circ h)(-3)$?
Answer (2)
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 final answer is 5 8 .
Explanation
Understanding the Problem We are given two functions: g ( x ) = x − 2 x + 1 and h ( x ) = 4 − x . We need to find the value of the composite function ( g ∘ h ) ( − 3 ) . The composite function ( g ∘ h ) ( x ) is defined as g ( h ( x )) .
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)) Next, we substitute the result from the previous step into g ( x ) to find g ( h ( − 3 )) , which is g ( 7 ) .
g ( 7 ) = 7 − 2 7 + 1 = 5 8 So, ( g ∘ h ) ( − 3 ) = g ( h ( − 3 )) = g ( 7 ) = 5 8 .
Final Answer Therefore, the value of ( g ∘ h ) ( − 3 ) is 5 8 .
Examples
Composite functions are used in real life to model situations where one function depends on another. For example, the cost of producing an item might depend on the number of items produced, and the number of items produced might depend on the number of employees working. The composite function would then give the cost of production as a function of the number of employees.
To find ( g ∘ h ) ( − 3 ) , first evaluate h ( − 3 ) to get 7 . Then substitute 7 into g ( x ) to find that g ( 7 ) = 5 8 . Thus, ( g ∘ h ) ( − 3 ) = 5 8 .
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