Determine whether each equation defines y as a function of x. 2x + y = 8

Verified step by step guidance

1. Rewrite the given equation in terms of y to isolate it. Start with the equation 2x + y = 8 and subtract 2x from both sides to get y = -2x + 8.
2. 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.
3. Examine the rewritten equation y = -2x + 8. Notice that for any given value of x, there is exactly one corresponding value of y because the equation is linear.
4. Conclude that the equation y = -2x + 8 defines y as a function of x because it satisfies the condition of having one unique output for each input.
5. If needed, you can also verify this by graphing the equation y = -2x + 8, which will produce a straight line, further confirming that it represents a function.

Key Concepts

Function Definition

A function is a relation where each input (x-value) corresponds to exactly one output (y-value). To determine if an equation defines y as a function of x, we must check if for every x, there is a unique y. This means that no x-value can produce multiple y-values.

Vertical Line Test

The vertical line test is a visual way 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 helps to quickly assess the uniqueness of y-values for given x-values.

Solving for y

To analyze whether an equation defines y as a function of x, we often rearrange the equation to solve for y. In the case of 2x + y = 8, isolating y gives y = 8 - 2x, which clearly shows that for each x, there is a corresponding unique y, confirming that it is a function.