Question

Determine the domain and range of the quadratic function. (Enter your answer using interval notation.)f(x) = 2x2 − 4x + 2domain range

Answer 1

we have the equation

`f\mleft(x\mright)=2x^2−4x+2`

The domain of any quadratic equation is all real numbers

so

The domain is the interval (-infinite, infinite)

To find out the range, we need the vertex

Convert the given equation into vertex form

`\begin{gathered} f\mleft(x\mright)=2x^2−4x+2 \\ f(x)=2(x^2-2x)+2 \end{gathered}`

Complete the square

`\begin{gathered} f(x)=2(x^2-2x+1-1)+2 \\ f(x)=2(x^2-2x+1)+2-2 \\ f(x)=2(x^2-2x+1) \\ f(x)=2(x-1)^2 \end{gathered}`

The vertex is the point (1,0)

The vertical parabola opens upward (the leading coefficient is positive)

The vertex is a minimum

therefore

The range is the interval [0, infinite)

All real numbers greater than or equal to zero