If [tex]f(x)=16 x-30[/tex] and [tex]g(x)=14 x-6[/tex], for which value of [tex]x[/tex] does [tex](f-g)(x)=0[/tex]?
Answer (2)
Define the difference: ( f − g ) ( x ) = f ( x ) − g ( x ) .
Substitute the given functions: ( f − g ) ( x ) = ( 16 x − 30 ) − ( 14 x − 6 ) .
Simplify the expression: ( f − g ) ( x ) = 2 x − 24 .
Solve for x when ( f − g ) ( x ) = 0 : x = 12 .
12
Explanation
Understanding the Problem We are given two functions, f ( x ) = 16 x − 30 and g ( x ) = 14 x − 6 . Our goal is to find the value of x for which ( f − g ) ( x ) = 0 . This means we need to find the x that makes the difference between the two functions equal to zero.
Finding the Difference First, let's find the expression for ( f − g ) ( x ) . This is simply f ( x ) − g ( x ) . So, we have: ( f − g ) ( x ) = f ( x ) − g ( x ) = ( 16 x − 30 ) − ( 14 x − 6 ) Now, we simplify the expression by combining like terms.
Simplifying the Expression Next, we simplify the expression: ( f − g ) ( x ) = 16 x − 30 − 14 x + 6 Combine the x terms: 16 x − 14 x = 2 x .
Combine the constant terms: − 30 + 6 = − 24 .
So, we have: ( f − g ) ( x ) = 2 x − 24
Solving for x Now, we set ( f − g ) ( x ) equal to zero and solve for x :
2 x − 24 = 0 Add 24 to both sides of the equation: 2 x = 24 Divide both sides by 2: x = 2 24 = 12
Final Answer Therefore, the value of x for which ( f − g ) ( x ) = 0 is x = 12 .
Examples
Imagine you are comparing the costs of two different phone plans. Plan A costs $30 per month plus $16 for each gigabyte of data you use. Plan B costs $6 per month plus $14 for each gigabyte of data you use. You want to find out how many gigabytes of data you need to use for the two plans to cost the same. This is exactly the problem we solved! The value of x represents the number of gigabytes, and we found that when you use 12 gigabytes, the two plans cost the same amount.
The value of x that makes the difference between the two functions ( f − g ) ( x ) equal to zero is 12 . We found this by simplifying the expression ( f − g ) ( x ) and solving for x . Thus, x = 12 .
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