Determine whether the equation defines y as a function of x
y=1/x
y is a function of x if you exclude x=0 from the domain.
Determine if the equation y^3+3x=5 defines y as a function of x
Well, I have to say, this equation is quite the joker! It's a classic case of a reciprocal function. For every value of x (except x=0, we don't want any division by zero shenanigans), there's a corresponding value of y. So, yes, I would say that y is indeed defined as a function of x in this equation.
To determine whether the equation y = 1/x defines y as a function of x, we need to check if there is a unique y-value associated with each x-value.
In this equation, y is defined in terms of x using the rule y = 1/x. To check if it is a function, we can examine if there are any x-values that cause multiple y-values or no y-value at all.
Let's consider a few examples to determine if the equation is a function:
1. For x = 0, plugging it into the equation gives y = 1/0. However, division by zero is undefined, so there is no y-value associated with x = 0.
2. For positive values of x (x > 0), as x increases, y decreases. Similarly, for negative values of x (x < 0), as x decreases, y also decreases. Hence, for every nonzero x-value, there is exactly one y-value.
Based on our analysis, we can conclude that the equation y = 1/x defines y as a function of x in the domain where x does not equal 0.