How do you find the domain and range of #y = x^2 - x + 5#?

Question

How do you find the domain and range of #y = x^2  - x + 5#?

Abigail Allen · Accepted Answer

Domain is all real numbers and the range is all real numbers #>=# #19/4#. The easiest way is to rewrite it as y= #(x-1/2)^2# +#19/4#.  This represents a parabola opening up with its vertex at ( #1/2#, #19/4#).  This clearly depicts the domain as all real numbers (-inf, inf) and range as all real numbers #>=# #19/4# that is [#19/4#, infy)

Nathan Axel · Answer

undefined To find the domain of the function y = x^2 - x + 5, we note that it is a quadratic function. Quadratic functions are defined for all real numbers. Therefore, the domain of this function is all real numbers.

To find the range of the function, we need to determine the minimum or maximum value of the quadratic function. Since the coefficient of x^2 is positive (1 in this case), the parabola opens upwards. The vertex of the parabola represents the minimum value of the function. The x-coordinate of the vertex can be found using the formula x = -b/2a, where a = 1 and b = -1 in this case. Plugging these values into the formula, we get x = 1/2.

To find the corresponding y-coordinate (minimum value), we substitute x = 1/2 into the function: y = (1/2)^2 - (1/2) + 5 = 1/4 - 1/2 + 5 = 25/4 - 2/4 + 20/4 = 43/4.

Therefore, the minimum value of the function occurs at x = 1/2 and y = 43/4. Since the parabola opens upwards, there is no maximum value, and the range of the function is all real numbers greater than or equal to 43/4.