Alexmahone said:Give an example of a function $\displaystyle f(x)$ for which $\displaystyle f([-1,\ 1])=(-\infty,\ \infty)$.
My thoughts: $\displaystyle f(x)=\frac{x}{(x-1)(x+1)}$ is a function for which $f((-1,\ 1))=(-\infty,\ \infty)$.
A function in mathematics is a rule or relationship between two quantities, often represented by the letters "x" and "y". It maps each input number (x) to a unique output number (y).
f(x) is a notation used to represent a function. The "x" inside the parentheses represents the input value, and the "f" outside the parentheses represents the name of the function. This notation allows us to define and work with different functions in a clear and organized manner.
One example of a function using f(x) is f(x) = 2x + 1. This function takes an input value "x" and multiplies it by 2, then adds 1 to the result to get the output value "y". For example, if we plug in x = 3, we get f(3) = 2(3) + 1 = 7. So, the output value for an input of 3 is 7.
To graph a function using f(x), we plot points on a coordinate plane where the x-values are the input values and the y-values are the output values. Then, we connect the points to create a line or curve, depending on the type of function. The resulting graph represents the relationship between the input and output values of the function.
Functions are a fundamental concept in mathematics and are used to model real-world situations, make predictions, and solve problems. Understanding functions allows us to analyze and interpret data, identify patterns, and make informed decisions. It is also a crucial building block for more advanced mathematical concepts such as calculus and differential equations.