Select the correct answer.
Which function has the same graph as [tex]$x+y=11$[/tex]?
A. [tex]$f(x)=-y+11$[/tex]
B. [tex]$f(x)=-x+11$[/tex]
C. [tex]$f(x)=x-11$[/tex]
D. [tex]$f(x)=y-11$[/tex]
Answer (2)
Start with the equation x + y = 11 .
Isolate y by subtracting x from both sides: y = 11 − x .
Express the equation as a function: f ( x ) = − x + 11 .
The correct answer is f ( x ) = − x + 11 .
Explanation
Understanding the Problem We are given the equation x + y = 11 and asked to find a function f ( x ) that has the same graph. This means we need to express y as a function of x .
Isolating y To find the function, we need to isolate y on one side of the equation. We can do this by subtracting x from both sides of the equation: x + y − x = 11 − x y = 11 − x
Expressing as a Function Now we can write this as a function f ( x ) :
f ( x ) = 11 − x This is the same as: f ( x ) = − x + 11
Selecting the Correct Option Now we compare our result, f ( x ) = − x + 11 , with the given options: A. f ( x ) = − y + 11 B. f ( x ) = − x + 11 C. f ( x ) = x − 11 D. f ( x ) = y − 11 Option B matches our result.
Final Answer Therefore, the function that has the same graph as x + y = 11 is f ( x ) = − x + 11 .
Examples
Understanding how to rewrite equations as functions is useful in many real-world scenarios. For example, if you have a budget of $11 and you want to buy apples and bananas, where x is the amount you spend on apples and y is the amount you spend on bananas, the equation x+y=11 represents your budget constraint. Rewriting this as y = 11-x allows you to easily determine how much you can spend on bananas if you know how much you're spending on apples.
The function that has the same graph as x + y = 11 is f ( x ) = − x + 11 . This is derived by isolating y in the original equation and rewriting it in function form. The correct answer is Option B: f ( x ) = − x + 11 .
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