Question

Find domain and range for f(x)=x^2-2x+2

Answer 1

Answer:

Domain: (-∞,∞) or  -∞ <$x$<∞

Range: [1,∞) or $y\geq 1$

Step-by-step explanation:

Given function:

$f(x)=x^2-2x+2$

To find range and domain of the function.

Solution:

The given function is a second degree function as the highest exponent of the variable is 2. Thus, it is a quadratic function.

For all quadratic functions the domain is a set of all real numbers. So, the domain can be given as: (-∞,∞) or  -∞ <$x$<∞

In order to find the range of the quadratic function, we will first determine if the function has a minimum point or the maximum point.

For a quadratic equation : $ax^2+bx+c$

1) If $a>0$ the function will have a minimum point.

2) If  $a$ the function will have a maximum point.

For the given function: $f(x)=x^2-2x+2$

$a=1$ which is greater than 0, and hence it will have a minimum point.

To find the range we will find the coordinates of the minimum point or the vertex of the function.

The x-coordinate $h$ of the vertex of a quadratic function is given by :

$h=\frac{-b}{2a}$

Thus, for the function:

$h=\frac{-(-2)}{2(1)}$

$h=\frac{2}{2}$  [Two negatives multiply to get a positive]

$h=1$

To find y-coordinate of the vertex can be found out by evaluating $f(h)$.

$k=f(h)$

Thus, for the function:

$k=f(1)=(1)^2-2(1)+2$

$k=f(1)=1-2+2$

$k=f(1)=1$

Thus, the minimum point of the function is at (1,1).

thus, range of the function is:

[1,∞) or $y\geq 1$