In Exercises 11–26, determine whether each equation defines y as a function of x. x² + y = 25

Verified step by step guidance

1. Step 1: Recall the definition of a function. A relation defines y as a function of x if, for every value of x, there is exactly one corresponding value of y.
2. Step 2: Start with the given equation: $x^2\; +\; y\; =\; 25$. Solve for y in terms of x by isolating y. Subtract $x^2$ from both sides to get $y\; =\; 25\; -\; x^2$.
3. Step 3: Analyze the resulting equation $y\; =\; 25\; -\; x^2$. For any given value of x, there is only one corresponding value of y because the equation does not involve any operations (like a square root) that could produce multiple outputs for the same input.
4. Step 4: Conclude that the equation $y\; =\; 25\; -\; x^2$ defines y as a function of x because it passes the vertical line test. This means that for every x-value, there is exactly one y-value.
5. Step 5: Verify your conclusion by considering the graph of the equation. The graph of $y\; =\; 25\; -\; x^2$ is a parabola that opens downward, and it satisfies the condition of being 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 crucial for determining if an equation defines y as a function of x.

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, then the graph does not represent a function. This test helps in analyzing equations to see if they can be expressed as y in terms 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. If the resulting expression for y can be expressed as a single output for each input x, then y is a function of x.