In Exercises 11–26, determine whether each equation defines y as a function of x. x + y = 16

Verified step by step guidance

1. Rewrite the given equation x + y = 16 in terms of y to determine if it can be expressed as a single output for each input x. Subtract x from both sides to isolate y: y = 16 - x.
2. Recall the definition of a function: A relation is a function if each input (x) corresponds to exactly one output (y).
3. Examine the rewritten equation y = 16 - x. For any value of x, there is exactly one corresponding value of y because the equation is linear and does not involve any operations (like squaring or taking a square root) that could produce multiple outputs.
4. Conclude that the equation y = 16 - x defines y as a function of x because it satisfies the condition of having one unique output for each input.
5. Optionally, you can verify this by graphing the equation y = 16 - x. The graph is a straight line, which passes the vertical line test, confirming that it is a function.

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 fundamental in determining if an equation defines one variable as a function of another.

Vertical Line Test

The vertical line test is a visual way to determine if a curve is a graph of a function. If any vertical line intersects the graph at more than one point, the relation is not a function. This test is particularly useful when analyzing graphs, but it can also be applied conceptually to equations to assess if they define one variable in terms of another.

Solving for y

To determine if an equation defines y as a function of x, one can solve the equation for y. If the resulting expression for y can be written as a single output for every input x, then y is a function of x. In the case of the equation x + y = 16, rearranging it to y = 16 - x allows us to see that for each x, there is a unique corresponding y.