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Which function has the same graph as $x+y=11$?


A$f(x)=-y+11$
B$f(x)=-x+11$
C$f(x)=x-11$
D$f(x)=y-11$

Verified Answer :

  • The equation \(x + y = 11\) represents a linear relationship between two variables \(x\) and \(y\).
  • To define this as a function \(f(x)\), we express \(y\) in terms of \(x\) by subtracting \(x\) from both sides, obtaining \(y = -x + 11\).
  • Comparing this to the given options, we identify that \(f(x) = -x + 11\) is the correct function.
  • The final answer is \(\boxed{f(x) = -x + 11}\).

Explanation

  1. Understanding the relationship between equations and functions

We are given the linear equation \(x + y = 11\) and asked to identify which function \(f(x)\) has the same graph. A graph of a function \(f(x)\) is defined as the set of all points \((x, y)\) such that \(y = f(x)\). Therefore, to express the given equation in the form of a function \(f(x)\), we must isolate the variable \(y\).

  1. Isolating the variable to define the function

Starting with the equation \(x + y = 11\), we subtract \(x\) from both sides of the equation to isolate \(y\):
\[y = 11 - x\]
This expression tells us that for any given \(x\), the corresponding \(y\) value on the graph is determined by the formula \(11 - x\). Therefore, the function that represents this graph is \(f(x) = 11 - x\), which can also be written as \(f(x) = -x + 11\).

  1. Comparing and selecting the correct function

Now we compare our derived function \(f(x) = -x + 11\) with the provided options:

  1. \(f(x) = -y + 11\)
  2. \(f(x) = -x + 11\)
  3. \(f(x) = x - 11\)
  4. \(f(x) = y - 11\)

The expression \(f(x) = -x + 11\) matches the second option. Thus, the function \(f(x) = -x + 11\) has the same graph as the linear equation \(x + y = 11\).

Examples

Understanding the relationship between equations and functions is essential in fields like physics and economics. For example, if the total cost of producing two types of items is fixed at \$\(\$1100\), represented by the equation \(x + y = 1100\), knowing the function \(f(x) = 1100 - x\) allows a manager to quickly calculate the cost of item \(y\) if they decide to produce a certain quantity of item \(x\). This helps in making rapid decisions about production budgets and resource allocation.

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