If [tex]f(x)=2x[/tex] and [tex]g(x)=5x+2[/tex], what is [tex](f \div g)(x)[/tex] when [tex]x=3[/tex]?
Answer (1)
Define the division of functions: ( f o b re ak o in d e n t ÷ g ) ( x ) = g ( x ) f ( x ) = 5 x + 2 2 x .
Substitute x = 3 into the expression: ( f o b re ak o in d e n t ÷ g ) ( 3 ) = 5 ( 3 ) + 2 2 ( 3 ) .
Simplify the expression: 15 + 2 6 = 17 6 .
The final answer is 17 6 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = 2 x and g ( x ) = 5 x + 2 . We need to find the value of ( f o b re ak o in d e n t ÷ g ) ( x ) when x = 3 . In other words, we need to evaluate g ( x ) f ( x ) at x = 3 .
Finding the Expression for (f/g)(x) First, let's find the expression for ( f o b re ak o in d e n t ÷ g ) ( x ) . This is simply g ( x ) f ( x ) . Substituting the given functions, we have ( f o b re ak o in d e n t ÷ g ) ( x ) = 5 x + 2 2 x .
Substituting x = 3 Now, we need to evaluate this expression at x = 3 . So, we substitute x = 3 into the expression: ( f o b re ak o in d e n t ÷ g ) ( 3 ) = 5 ( 3 ) + 2 2 ( 3 ) .
Simplifying the Expression Next, we simplify the expression: ( f o b re ak o in d e n t ÷ g ) ( 3 ) = 15 + 2 6 = 17 6 .
Final Answer Therefore, ( f o b re ak o in d e n t ÷ g ) ( 3 ) = 17 6 .
Examples
Imagine you are baking a cake and need to adjust the recipe based on the number of guests. If f ( x ) represents the amount of flour needed and g ( x ) represents the amount of sugar needed, where x is the number of guests, then ( f o b re ak o in d e n t ÷ g ) ( x ) tells you the ratio of flour to sugar. Evaluating this at x = 3 tells you the specific ratio needed for 3 guests. Understanding function division helps in scaling recipes, adjusting chemical mixtures, or even calculating financial ratios based on different variables.
