Polynomial functions: constant, affine and quadratic

Determine the domain of the following functions, their image, and in the case of a parabola determine the vertex:

  1. $f(x)=2x-3$
  2. $f(x)=-1$
  3. $f(x)=-x^2+4x-1$

  1. The function is affine. (Odd) the degree of the polynomial is $1$. Therefore, $Dom (f) = Im (f) = \mathbb{R}$.

  2. The function is constant. Therefore, $Dom (f) = \mathbb{R}$, $Im (f) =-1$.

  3. The function is a polynomial of degree $2$. Therefore its domain is $Dom (f) =\mathbb{R}$. To calculate the image first we must look for the vertex:

$$\Big(-\dfrac{b}{2a}, -\dfrac{b^2-4ac}{4a}\Big)=\Big(-\dfrac{4}{-2}, -\dfrac{16-4\cdot(-1)\cdot(-1)}{-4}\Big)=(2,3) $$

Since $a < 0$, the parabola goes down and therefore, $Im (f) = (-\infty, 3]$.

  1. $Dom (f) = Im (f) = \mathbb{R}$

  2. $Dom (f) = \mathbb{R}$, $Im (f) =-1$

  3. $Dom (f) = \mathbb{R}$, $Im (f) = (-\infty, 3]$, $v=(2,3)$

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