Question

What is the domain and range of the function `y = x^2- x + 5`?

Answer 1

Domain: `(-oo, oo)` or all reals
Range: `[19/4, oo)` or `" "y >= 19/4`

Explanation:

Given: `y = x^2 - x + 5`

The domain of an equation is usually `(-oo, oo)` or all reals unless there is a radical (square root) or a denominator (causes asymptotes or holes).

Since this equation is a quadratic (parabola), you would need to find the vertex. The vertex's `y`-value will be the minimum range or the maximum range if the equation is an inverted parabola (when the leading coefficient is negative).

If the equation is in the form: `Ax^2 +Bx +C = 0` you can find the vertex:

vertex: `(-B/(2A), f(-B/(2A)))`

For the given equation: `A = 1, B = -1, C = 5`

`-B/(2A) = 1/2`

`f(1/2) = (1/2)^2 - 1/2 + 5`

`f(1/2) = 1/4 - 2/4 + 20/4`

`f(1/2) = 19/4 = 4.75`

Domain: `(-oo, oo)` or all reals
Range: `[19/4, oo)` or `" "y >= 19/4`

graph{x^2-x+5 [-25.66, 25.66, -12.82, 12.83]}