Determine whether each equation defines y as a function of x. y = √x +4

Verified step by step guidance

1. Step 1: Recall the definition of a function. A function is a relation where each input (x) corresponds to exactly one output (y).
2. Step 2: Analyze the given equation y = √x + 4. The square root function (√x) is defined only for x ≥ 0, as the square root of a negative number is not a real number.
3. Step 3: For each valid input x (x ≥ 0), the square root function produces exactly one output. Adding 4 to this output does not change the fact that there is only one output for each input.
4. Step 4: Since the equation y = √x + 4 produces exactly one value of y for each valid x, it satisfies the definition of a function.
5. Step 5: Conclude that the equation y = √x + 4 defines y as a function of x.

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 function. If any vertical line intersects the graph of the relation at more than one point, then the relation is not a function. This test helps to quickly assess whether an equation like y = √x + 4 defines y as a function of x.

Square Root Function

The square root function, represented as y = √x, is defined only for non-negative values of x, meaning x must be greater than or equal to zero. This function produces a unique output for each valid input, reinforcing the idea that it defines y as a function of x. Understanding the domain of this function is essential for analyzing the given equation.