$f(x)=5 x^3$ and $g(x)=x+1$, find $(f ">• g)(x)$.
A. $5 x^3+1$
B. $5 x^4+5 x^3$
C. $5 x^4+1$
D. $6 x^3
Answer (2)
We are given f ( x ) = 5 x 3 and g ( x ) = x + 1 . To find • g)(x)"> ( f " > • g ) ( x ) , we need to find f ( g ( x )) .
First, we find g ( x ) = x + 1 .
Next, we substitute g ( x ) into f ( x ) to get f ( g ( x )) = f ( x + 1 ) = 5 ( x + 1 ) 3 .
Then, we expand the expression: 5 ( x + 1 ) 3 = 5 ( x 3 + 3 x 2 + 3 x + 1 ) = 5 x 3 + 15 x 2 + 15 x + 5 .
The final answer is 5 x 3 + 15 x 2 + 15 x + 5 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = 5 x 3 and g ( x ) = x + 1 . We need to find the composite function • g)(x)"> ( f " > • g ) ( x ) , which means we need to find f ( g ( x )) .
Finding g(x) First, we find g ( x ) , which is given as g ( x ) = x + 1 .
Substituting g(x) into f(x) Next, we substitute g ( x ) into f ( x ) to find f ( g ( x )) . This means we need to compute f ( x + 1 ) . So, we replace x in f ( x ) with ( x + 1 ) : f ( g ( x )) = f ( x + 1 ) = 5 ( x + 1 ) 3
Expanding the Expression Now, we expand and simplify the expression 5 ( x + 1 ) 3 . Recall that ( a + b ) 3 = a 3 + 3 a 2 b + 3 a b 2 + b 3 . Therefore, ( x + 1 ) 3 = x 3 + 3 x 2 ( 1 ) + 3 x ( 1 ) 2 + 1 3 = x 3 + 3 x 2 + 3 x + 1 So, 5 ( x + 1 ) 3 = 5 ( x 3 + 3 x 2 + 3 x + 1 ) = 5 x 3 + 15 x 2 + 15 x + 5
Comparing with Options Finally, we compare the result 5 x 3 + 15 x 2 + 15 x + 5 with the given options. None of the options match our result. However, let's re-examine the problem statement and the given options. It seems there might be a typo in the options. The correct expansion of f ( g ( x )) is 5 x 3 + 15 x 2 + 15 x + 5 .
Examples
Composite functions are used in various real-life scenarios. For example, consider a store that offers a discount of 10% on all items and then applies a sales tax of 5%. If x is the original price, the discounted price is g ( x ) = 0.9 x , and the price after tax is f ( x ) = 1.05 x . The final price is the composite function f ( g ( x )) = 1.05 ( 0.9 x ) = 0.945 x . This shows how composite functions can model sequential operations.
To find ( f ∘ g ) ( x ) , we substitute g ( x ) into f ( x ) resulting in 5 ( x + 1 ) 3 . Expanding this gives us 5 x 3 + 15 x 2 + 15 x + 5 , which is not listed as an option. The correct answer is 5 x 3 + 15 x 2 + 15 x + 5 .
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