Question

Which of the following relations are also functions? Explain. $$y=3 x^{2}-5$$

Short answer

The relation \(y=3x^{2}-5\) is a function because for every x-value, there is only one corresponding y-value as determined by the equation, which is characteristic of quadratic functions.

Step-by-step solution

Identify the Relation

First, understand that the relation given is a mathematical expression that connects the variable y with the variable x through a specific rule or equation.

Define a Function

A function is a type of relation where each input (often x-value) is associated with exactly one output (y-value). This means that for every x-value, there can only be one corresponding y-value.

Apply the Vertical Line Test

One way to determine if a relation is a function is by using the vertical line test. If a vertical line drawn at any x-value on the graph of the relation intersects the graph at exactly one point, the relation is a function.

Verify the Relation as a Function

Since the given relation is a quadratic equation of the form y=ax^2+bx+c and such equations always graph as parabolas that open upwards or downwards, they will pass the vertical line test. Therefore, this relation is a function because for every x-value there is one unique y-value determined by the equation.

Key concepts

Vertical Line Test

Understanding whether a given relation qualifies as a function is crucial in mathematics. One common method to ascertain this is via the vertical line test. This test is a visual way of determining the functionhood of a relation by graphing the relation and observing its intersections with vertical lines.

If any vertical line crosses the graph more than once, then the relation is not a function, as it would violate the rule that every x-value must correspond to exactly one y-value. In contrast, if every vertical line intersects the graph at only one point, it satisfies the definition of a function. This is because it demonstrates that each input has a single unique output. For the quadratic equation given in the exercise, graphing it yields a parabola – a U-shaped curve that opens either upwards or downwards – which will never be intersected more than once by any vertical line, hence verifying that it is indeed a function.

Quadratic Equations

Quadratic equations form an essential part of algebra and appear frequently across various fields of mathematics. A quadratic equation is typically expressed in the standard form of \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants with \( a \) not equal to zero. This type of equation generates a specific curve on the graph known as a parabola.

Characteristically, parabolas have a symmetrical shape, and they open upwards if \( a > 0 \) and downwards if \( a < 0 \). The highest or lowest point of a parabola is called the vertex, which is a significant feature as it provides key insights into the graph's properties. Quadratic equations are vital because they model various real-world scenarios, such as projectile motion and the design of certain types of lenses.

Function Definition

A function is a fundamental concept in algebra and higher-level mathematics. Formally, it is a relation between two sets that assigns to each element of the first set (the domain) exactly one element of the second set (the range). In more intuitive terms, a function can be thought of as a 'machine' or 'rule': you input a number (from the domain), and the function processes it to produce an output (in the range).

Each function has its own formula or rule, such as \( y = 3x^2 - 5 \) from the exercise, which mathematically describes how to calculate the output for any given input. The graphical representation of a function is a set of points that maps every x-value to a y-value. Understanding functions is key since they describe the relationships between variable quantities and allow for predictions and analysis of the natural and manmade phenomena.