For [tex]f(x)=x+3[/tex] and [tex]g(x)=5x+3[/tex], find the following functions.
a. [tex](f \circ g)(x)[/tex];
b. [tex](g \circ f)(x)[/tex];
c. [tex](f \circ g)(1)[/tex];
d. [tex](g \circ f)(1)[/tex]
Answer (2)
First, find the composite function ( f ∘ g ) ( x ) by substituting g ( x ) into f ( x ) , resulting in 5 x + 6 .
Next, find the composite function ( g ∘ f ) ( x ) by substituting f ( x ) into g ( x ) , resulting in 5 x + 18 .
Evaluate ( f ∘ g ) ( 1 ) by substituting x = 1 into 5 x + 6 , which gives 11 .
Evaluate ( g ∘ f ) ( 1 ) by substituting x = 1 into 5 x + 18 , which gives 23 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = x + 3 and g ( x ) = 5 x + 3 . We need to find the composite functions ( f ∘ g ) ( x ) and ( g ∘ f ) ( x ) , and then evaluate these composite functions at x = 1 . The composition ( f ∘ g ) ( x ) means f ( g ( x )) , and ( g ∘ f ) ( x ) means g ( f ( x )) .
Finding ( f c i rc g ) ( x ) First, we find ( f ∘ g ) ( x ) = f ( g ( x )) . We substitute g ( x ) into f ( x ) : f ( g ( x )) = f ( 5 x + 3 ) = ( 5 x + 3 ) + 3 = 5 x + 6.
Finding ( g c i rc f ) ( x ) Next, we find ( g ∘ f ) ( x ) = g ( f ( x )) . We substitute f ( x ) into g ( x ) : g ( f ( x )) = g ( x + 3 ) = 5 ( x + 3 ) + 3 = 5 x + 15 + 3 = 5 x + 18.
Finding ( f c i rc g ) ( 1 ) Now, we find ( f ∘ g ) ( 1 ) . We substitute x = 1 into the expression we obtained for ( f ∘ g ) ( x ) : ( f ∘ g ) ( 1 ) = 5 ( 1 ) + 6 = 5 + 6 = 11.
Finding ( g c i rc f ) ( 1 ) Finally, we find ( g ∘ f ) ( 1 ) . We substitute x = 1 into the expression we obtained for ( g ∘ f ) ( x ) : ( g ∘ f ) ( 1 ) = 5 ( 1 ) + 18 = 5 + 18 = 23.
Examples
Composite functions are used in various real-life scenarios. For example, consider a store that marks up the price of an item by a certain percentage, and then applies a discount. If f ( x ) represents the markup function and g ( x ) represents the discount function, then the composite function ( g ∘ f ) ( x ) represents the final price of the item after both the markup and the discount are applied. Understanding composite functions helps in analyzing such situations and predicting outcomes.
The composite functions are ( f ∘ g ) ( x ) = 5 x + 6 and ( g ∘ f ) ( x ) = 5 x + 18 . The evaluated results at x = 1 are ( f ∘ g ) ( 1 ) = 11 and ( g ∘ f ) ( 1 ) = 23 . Thus, the answers are 11 and 23 respectively.
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