Problem 137 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Whether \(g=\\{(1,1),(2,3),(3,5),(4,7)\\}\) a function or not? If \(g\) is defined by the rule \(g(x)=a x+b\), then what values should be assigned to \(a\) and \(b\) ?

Step-by-Step Solution

Answer

The given set of points \(\\{(1,1),(2,3),(3,5),(4,7)\\}\) represents a function, as each x-value has a unique y-value. By solving the simultaneous equations, we find the coefficients a and b for the linear function g(x) = ax + b. The values are a = 2 and b = -1, so the function is defined by the rule \(g(x) = 2x - 1\).

Step 1: Verify if it is a function

To check if the given set of points is a function or not, we need to see if each input (x-value) has a unique output (y-value). The given points are: \(\\{(1,1),(2,3),(3,5),(4,7)\\}\) We can see that all the x-values (1, 2, 3, and 4) are unique, and each x-value corresponds to one and only one y-value: 1->1, 2->3, 3->5, and 4->7. Since each x-value (input) has a unique y-value (output), the given set of points can represent a function.

Step 2: Compute the coefficients a and b for the linear function

Now that we know the given set of points defines a function, we can proceed to finding the values of a and b in the linear function, g(x) = ax + b. We have the following four points: \(\\{(1,1),(2,3),(3,5),(4,7)\\}\). Let's use the first two points to derive two simultaneous equations involving a and b. For point (1,1): \(g(1) = a(1) + b = 1\) For point (2,3): \(g(2) = a(2) + b = 3\) Now we have two equations: 1. \(a + b = 1\) 2. \(2a + b = 3\)

Step 3: Solve the simultaneous equations

To solve the simultaneous equations: 1. \(a + b = 1\) 2. \(2a + b = 3\) We can eliminate b from both equations by subtracting equation 1 from equation 2: \(2a + b - (a + b) = 3 - 1\) This simplifies to: \(a = 2\) Now substitute the value of a into equation 1: \(2 + b = 1\) This simplifies to: \(b = -1\) Now we have values for a and b: a = 2 b = -1

Step 4: Write the function g(x)

Now that we have found the values of a and b, we can write the function g(x): \(g(x) = 2x - 1\) So, g is a function defined by the rule \(g(x) = 2x - 1\) with values a = 2 and b = -1.

Key Concepts

Linear Functions

In mathematics, linear functions are a specific type of function with a standard form represented as \( f(x) = ax + b \). These functions graph a straight line, and their most essential feature is that the rate of change between any two points on the line is constant. This is known as the slope, symbolized by \( a \).

• **Constant Change**: The change in \( y \) given a change in \( x \) is always the same.
• **Intercept**: The value \( b \) represents where the line crosses the y-axis.

Linear functions are particularly helpful for modeling real-world relationships that change at a steady rate. Examples include calculating total costs given a fixed price per unit or predicting distance traveled over constant speed.

Simultaneous Equations

To determine the exact equation of a linear function from a set of points, it may be necessary to use simultaneous equations. These are multiple equations solved together to find common variables. This method is helpful when a function is represented by unknown coefficients, like \( a \) and \( b \), and provides a system to solve these unknowns.

• **Formulating Equations**: Based on known points, you set up equations by substituting the x and y values into our linear function equation.
• **Elimination or Substitution**: These are common methods to find solutions, where variables are eliminated to solve the remaining equation.

By introducing two points from the function, we derive two equations, enabling us to solve for both the slope and the y-intercept, which define the function entirely.

Function Representation

A function in mathematics is a rule or relationship that maps elements from one set (inputs) to elements in another set (outputs) such that each input is related to exactly one output. Functions can be of various forms in mathematics, but understanding their representation is key.

• **Set of Points**: One method to represent a function is through a set of ordered pairs, each comprising an input and its corresponding output.
• **Equation Form**: Functions can often be expressed using an equation, describing the relationship between variables.

Graphical representation is also common, where the relation is shown as a curve or line, assisting visual comprehension of how input values translate to output values.

Unique Outputs

An essential characteristic of functions is that each input produces one unique output. This means, for every value you 'put into' a function, you get just one 'result.'

• **Definition Compliance**: The requirement of one output per input ensures the definition of a function is met.
• **Real-World Relevance**: This characteristic is crucial in real-world problems where consistency is needed; for instance, one height value for each person.

In the context of examining a set of points or plotting a function, confirming that each \( x \)-value is paired with only one \( y \)-value assures us that the dataset or equation indeed represents a function.