Determine whether the equation defines y as a function of x. [tex]y=\frac{4}{x+3}[/tex] Does the equation define [tex]y[/tex] as a function of [tex]x[/tex]? Yes No
Answer (2)
Determine the domain of the function: x = − 3 .
Verify that for each x in the domain, there is only one corresponding y value.
Conclude that the equation defines y as a function of x .
The equation defines y as a function of x : Y es
Explanation
Understanding the Function Definition To determine whether the equation y = x + 3 4 defines y as a function of x , we need to check if for every value of x in the domain, there is only one corresponding value of y .
Finding the Domain The domain of the function is all real numbers except for the value of x that makes the denominator equal to zero. We need to find that value: x + 3 = 0 x = − 3
So, x can be any real number except − 3 .
Analyzing the Uniqueness of y For any x = − 3 , the expression x + 3 will be a non-zero number. Therefore, x + 3 4 will result in a unique real number for y . This means that for every valid x , there is only one corresponding y value.
Conclusion Since for every x in the domain (all real numbers except − 3 ), there is exactly one value of y , the equation y = x + 3 4 defines y as a function of x .
Examples
In real life, this type of function can model various inverse relationships. For example, the time it takes to complete a task ( y ) can be related to the number of people working on it ( x ). If y = x + 3 4 , it means there's a baseline time of 3 units even without anyone working, and the remaining time decreases as more people join in. Understanding such functions helps in resource allocation and predicting outcomes based on changing conditions.
The equation y = x + 3 4 defines y as a function of x because there is only one unique value of y for every acceptable value of x . The domain excludes x = − 3 , ensuring that all other values of x yield a unique result for y . Therefore, the answer is: Yes.
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