In Exercises 11–26, determine whether each equation defines y as a function of x. xy - 5y =1

Verified step by step guidance

1. Rewrite the given equation to isolate y. Start with the equation: $xy\; -\; 5y\; =\; 1$. Factor out y from the left-hand side: $y(x\; -\; 5)\; =\; 1$.
2. Divide both sides of the equation by $(x\; -\; 5)$ to solve for y: $y\; =\; \backslash frac\{1\}\{x\; -\; 5\}$. This expresses y explicitly in terms of x.
3. Recall the definition of a function: For y to be a function of x, each input value of x must correspond to exactly one output value of y.
4. Examine the equation $y\; =\; \backslash frac\{1\}\{x\; -\; 5\}$. For any value of x except $x\; =\; 5$ (where the denominator becomes zero), there is exactly one value of y for each x.
5. Conclude that the equation defines y as a function of x, except at $x\; =\; 5$, where the function is undefined due to division by zero.

Key Concepts

Function Definition

A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. In mathematical terms, for a relation to be a function, no two ordered pairs can have the same first element with different second elements. This concept is crucial for determining if an equation defines y as a function of x.

Solving for y

To determine if an equation defines y as a function of x, it is often necessary to solve the equation for y. This involves isolating y on one side of the equation, which allows us to express y explicitly in terms of x. If we can express y as a single output for each input x, then y is a function of x.

Vertical Line Test

The vertical line test is a visual method used to determine if a graph represents a function. If any vertical line intersects the graph at more than one point, the relation is not a function. This test can be applied after graphing the equation derived from the original equation to confirm whether y is indeed a function of x.