Consider the function represented by the equation $x-y=3$. What is the equation written in function notation, with $x$ as the independent variable?
A. $f(x)=y+3$
B. $f(x)=-y-3$
C. $f(x)=-x+3$
D. $f(x)=x-3$
Answer (2)
Isolate y in the equation x − y = 3 .
Subtract x from both sides: − y = 3 − x .
Multiply both sides by − 1 : y = x − 3 .
Express the equation in function notation: f ( x ) = x − 3 . The final answer is x − 3 .
Explanation
Understanding the Problem We are given the equation x − y = 3 and we want to express it in function notation, f ( x ) , where x is the independent variable. This means we need to solve for y in terms of x .
Isolating y To solve for y , we can start by isolating y on one side of the equation. We have
x − y = 3
Subtracting x Subtracting x from both sides gives
− y = 3 − x
Multiplying by -1 Multiplying both sides by − 1 gives
y = x − 3
Function Notation Now we can write the equation in function notation by replacing y with f ( x ) :
f ( x ) = x − 3
Final Answer Therefore, the equation in function notation is f ( x ) = x − 3 .
Examples
In real life, this type of function can be used to model scenarios where there is a fixed relationship between two variables. For example, if you are selling items for x dollars each and it costs you a fixed 3 t o p ro d u cee a c hi t e m , t h e n yo u r p ro f i t f(x) c anb ere p rese n t e d a s f(x) = x - 3$. This means your profit is the selling price minus the cost of production. Understanding functions helps in predicting outcomes based on different inputs.
To write the equation x − y = 3 in function notation, we solve for y to get y = x − 3 . Therefore, we express this as f ( x ) = x − 3 . The correct option is D: f ( x ) = x − 3 .
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