Determine whether the equation defines y as a function of x y=4/x+4
since there is only one value of y for each value of x, it is a function.
These things are called "rational functions" for a reason.
To determine whether the equation defines y as a function of x, we need to check if for every value of x, there is exactly one corresponding value of y.
In the given equation, y = 4/x + 4, we can rewrite it as y = 4/(x+4).
If we consider different values of x, we can see that for each value of x, there is exactly one corresponding value of y. Therefore, yes, the equation defines y as a function of x.
To determine if the equation defines y as a function of x, we need to check if there is a unique output value of y for every input value of x.
The given equation is y = 4/x + 4.
To confirm if this equation defines y as a function of x, we need to ensure that for every x-value we input, we get a unique corresponding y-value.
In this case, since there are no restrictions or conditions given, we can assume that any real number can be used for x.
To determine if the equation passes the vertical line test, which checks for uniqueness, we need to ensure that no vertical line intersects the graph of the equation in more than one point.
Taking the given equation: y = 4/x + 4, we can rewrite it as y = 4(1/x) + 4.
Now, let's analyze the behavior of the equation:
- As x approaches positive infinity, 1/x approaches zero, and thus y approaches 4(0) + 4 = 4.
- As x approaches negative infinity, 1/x approaches zero, and again, y approaches 4(0) + 4 = 4.
Based on this analysis, we see that for any value of x, the expression 4/x is never equal to zero (except when x is zero, which we will address separately). This implies that for any input value of x (excluding zero), the equation y = 4/x + 4 will provide a unique output value of y.
However, the equation does not define y as a function of x when x = 0 because division by zero is undefined. Therefore, the equation y = 4/x + 4 does not define y as a function of x since we have a singularity at x = 0.
In summary, the equation y = 4/x + 4 defines y as a function of x for all x-values except x = 0.