Find $( h \circ g )( x )$ for the indicated functions.
$h(x)=\sqrt{x+7}, g(x)=\\frac{3}{x+5}$
$(h \circ g)(x) =$
Answer (2)
Substitute g ( x ) into h ( x ) : h ( g ( x )) = g ( x ) + 7 .
Substitute g ( x ) = x + 5 3 : h ( g ( x )) = x + 5 3 + 7 .
Simplify the expression inside the square root: x + 5 3 + 7 ( x + 5 ) = x + 5 7 x + 38 .
The final composite function is: x + 5 7 x + 38 .
Explanation
Understanding the Problem We are given two functions, h ( x ) = x + 7 and g ( x ) = x + 5 3 . Our goal is to find the composite function ( h ∘ g ) ( x ) , which means we need to find h ( g ( x )) . In other words, we need to substitute g ( x ) into h ( x ) wherever we see x .
Substituting g(x) into h(x) To find h ( g ( x )) , we replace x in h ( x ) with g ( x ) . So we have h ( g ( x )) = g ( x ) + 7 Now we substitute g ( x ) = x + 5 3 into the expression: h ( g ( x )) = x + 5 3 + 7
Simplifying the Expression To simplify the expression inside the square root, we need to find a common denominator. The common denominator is x + 5 . So we rewrite 7 as x + 5 7 ( x + 5 ) :
h ( g ( x )) = x + 5 3 + x + 5 7 ( x + 5 ) Now we can combine the fractions: h ( g ( x )) = x + 5 3 + 7 ( x + 5 ) Distribute the 7: h ( g ( x )) = x + 5 3 + 7 x + 35 Combine like terms: h ( g ( x )) = x + 5 7 x + 38
Final Answer Therefore, the composite function ( h ∘ g ) ( x ) is: ( h ∘ g ) ( x ) = x + 5 7 x + 38
Examples
Composite functions are useful in many real-world scenarios. For example, consider a store that is offering a discount on all of its items. Let g ( x ) be the price of an item after a discount of 10%, and let h ( x ) be the amount you pay after sales tax of 8%. Then ( h ∘ g ) ( x ) represents the final price you pay for an item after both the discount and sales tax are applied. Understanding composite functions helps you to model and analyze such situations.
To find the composite function ( h ∘ g ) ( x ) , we substitute g ( x ) into h ( x ) leading to the expression x + 5 7 x + 38 . This showcases the transformation between the functions. The final result for the composite function is ( h ∘ g ) ( x ) = x + 5 7 x + 38 .
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