Insert the composite functions f(g) and g(f), when g = sin(x) and f = 5x^3 + 5. Insert an algebraic expression for composite function fg and gf, using scalar x and simplify your an... Insert the composite functions f(g) and g(f), when g = sin(x) and f = 5x^3 + 5. Insert an algebraic expression for composite function fg and gf, using scalar x and simplify your answers to include sin(x).

Question image

Understand the Problem

The question is asking to find and insert algebraic expressions for the composite functions f(g(x)) and g(f(x)) when g and f are given functions. It also requires simplifying the answers to include sin(x).

Answer

- For \( f(g(x)) \): \( 5\sin(x)^3 + 5 \) - For \( g(f(x)) \): \( \sin(5x^3 + 5) \)
Answer for screen readers
  1. For ( f(g(x)) ): ( 5\sin(x)^3 + 5 )

  2. For ( g(f(x)) ): ( \sin(5x^3 + 5) )

Steps to Solve

  1. Identify the Functions

We have two functions:

  • ( g(x) = \sin(x) )
  • ( f(x) = 5x^3 + 5 )
  1. Find ( f(g(x)) )

This means we need to substitute ( g(x) ) into ( f(x) ):

$$ f(g(x)) = f(\sin(x)) = 5(\sin(x))^3 + 5 $$

  1. Simplify ( f(g(x)) )

The expression is already simplified as ( 5\sin(x)^3 + 5 ), which includes ( \sin(x) ).

  1. Find ( g(f(x)) )

Now we substitute ( f(x) ) into ( g(x) ):

$$ g(f(x)) = g(5x^3 + 5) = \sin(5x^3 + 5) $$

  1. Simplify ( g(f(x)) )

The expression can't be simplified further, so we have:

$$ g(f(x)) = \sin(5x^3 + 5) $$

  1. For ( f(g(x)) ): ( 5\sin(x)^3 + 5 )

  2. For ( g(f(x)) ): ( \sin(5x^3 + 5) )

More Information

The composite function ( f(g(x)) ) shows how ( g(x) ), which gives the sine of ( x ), is used in ( f(x) ). Similarly, ( g(f(x)) ) applies the sine function to a cubic polynomial, revealing how composite functions can change the output significantly.

Tips

  • Forgetting to substitute the inner function completely. Make sure to plug all parts of ( g(x) ) into ( f(x) ) and vice versa.
  • Not simplifying fully. Always check to see if the expression can be simplified or rewritten using the required function form.
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