Understanding the Domain of y x^2 - 4x 5

Understanding the domain of a function is a fundamental concept in mathematics. The domain of a function is the set of all values of the independent variable (typically x) for which the function is defined. In this article, we will explore the domain of the function y x^2 - 4x 5.

What is the Domain of a Function?

The domain of a function is the set of all values of the independent variable, x, for which the function is defined and produces a real and valid result. To find the domain, we need to determine what values of x can be plugged into the function and still yield a valid output (y).

Determining the Domain of y x^2 - 4x 5

When dealing with the function y x^2 - 4x 5, the first step is to consider whether any restrictions exist on the value of x. In this case, the function is a polynomial, which means it is defined for all real numbers. Therefore, the value of x can be any real number, and the function will produce a defined and valid output (y).

Let's illustrate this with a few examples:

y (-2)^2 - 4(-2) 5 4 8 5 17 For x 2, we have

y 2^2 - 4(2) 5 4 - 8 5 1 For x 0, we have

y 0^2 - 4(0) 5 5

As these examples show, the function is defined for all real values of x, with no restrictions. Therefore, the domain of the function y x^2 - 4x 5 is all real numbers, denoted as R.

How to Find the Domain Using Inverse Functions

To find the domain of a function, you can sometimes use its inverse. The idea is to swap the dependent and independent variables, solve for the new dependent variable, and identify the domain of this inverse function. The domain of the inverse function will correspond to the range of the original function.

For the function y x^2 - 4x 5, the corresponding inverse function after swapping x and y and solving for y is:

This equation can be rewritten in the standard quadratic form x (y^2 - 4y 4) 1 which can be further simplified as

x (y-2)^2 1

Solving for y, we get

y 2 ± √(x - 1)

The domain of this inverse function is all real values of x where the expression under the square root is non-negative. This means x - 1 ≥ 0 or x ≥ 1. However, the domain of the original function is all real numbers, indicating that the root function provides the range of the original function.

The Axis of Symmetry

For a quadratic function of the form y ax^2 bx c, the axis of symmetry is a vertical line that divides the parabola into two symmetrical halves. The formula for the axis of symmetry is x -b/(2a).

For the function y x^2 - 4x 5, we have a 1 and b -4. Plugging these values into the formula gives us:

x -(-4)/(2 * 1) 4/2 2

Therefore, the axis of symmetry for the function y x^2 - 4x 5 is the vertical line x 2.

Conclusion

The domain of the function y x^2 - 4x 5 is all real numbers. This is because it is a polynomial function, which is defined for every real value of x. Understanding the domain and range of functions is crucial in mathematics and helps in analyzing the behavior of functions. Additionally, knowing the axis of symmetry provides insights into the shape and position of the quadratic function.