$f(x)=4x+8, f(a+2)=12$. $a=?$
Answer (1)
∙ Substitute a + 2 into the function: f ( a + 2 ) = 4 ( a + 2 ) + 8 .
∙ Set the expression equal to 12: 4 ( a + 2 ) + 8 = 12 .
∙ Solve the equation for a : 4 a + 16 = 12 ⇒ 4 a = − 4 ⇒ a = − 1 .
∙ The value of a is − 1 .
Explanation
Understanding the Problem We are given the function f ( x ) = 4 x + 8 and the condition f ( a + 2 ) = 12 . Our goal is to find the value of a .
Substituting into the Function First, we substitute a + 2 into the function f ( x ) :
f ( a + 2 ) = 4 ( a + 2 ) + 8
Setting up the Equation We are given that f ( a + 2 ) = 12 , so we set the expression equal to 12:
4 ( a + 2 ) + 8 = 12
Distributing Now, we solve for a . Distribute the 4:
4 a + 8 + 8 = 12
Combining Like Terms Combine like terms:
4 a + 16 = 12
Isolating the Term with a Subtract 16 from both sides:
4 a = 12 − 16
4 a = − 4
Solving for a Divide both sides by 4:
a = 4 − 4
a = − 1
Final Answer Therefore, the value of a is -1.
Examples
In real life, functions like this can model costs or values that change linearly. For example, imagine a phone plan where you pay a fixed monthly fee plus a charge per text message. If x is the number of text messages and f ( x ) is your total bill, the equation f ( x ) = 4 x + 8 could represent a plan with a fixed fee of $8 and a charge of $4 per text. If your bill is $12, you can use the equation to find out how many texts you sent. Similarly, this type of problem can be used to calculate the number of items sold to reach a certain revenue target, given a fixed cost and a profit per item.
