# What is the range of the function F(X) = ( X - 1)^2 + 6?

Aug 20, 2017

All real numbers $Y$ such that $Y \ge 6$

#### Explanation:

The range of a function $F \left(X\right)$ is the set of all numbers that can be produced by the function.

Calculus gives you some better tools to answer this type of equation, but since it's algebra, we won't use them. In this case, the best tool is probably to graph the equation.

It is of quadratic form, so the graph is a parabola, opening up.

This means that it has a minimum point. This is at $X = 1$, at which

$F \left(X\right) = 6$

There is NO value of $X$ for which the function produces a result less than $6$.

Therefore the range of the function is all real numbers $Y$ such that

$Y \ge 6$

Aug 20, 2017

$\left[6 , \infty\right) .$

#### Explanation:

Observe that, $\forall x \in \mathbb{R} , {\left(x - 1\right)}^{2} \ge 0.$

Adding $6 , {\left(x - 1\right)}^{2} + 6 \ge 0 = 6 = 6.$

$\therefore \forall x \in \mathbb{R} , f \left(x\right) \ge 6.$

Hence, $\text{the Range of f=} \left[6 , \infty\right) .$