Determine the domain of the following functions, their image, and in the case of a parabola determine the vertex:
- $$f(x)=2x-3$$
- $$f(x)=-1$$
- $$f(x)=-x^2+4x-1$$
See development and solution
Development:
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The function is affine. (Odd) the degree of the polynomial is $$1$$. Therefore, $$Dom (f) = Im (f) = \mathbb{R}$$.
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The function is constant. Therefore, $$Dom (f) = \mathbb{R}$$, $$Im (f) =-1$$.
- 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, $$Im (f) = (-\infty, 3]$$.
Solution:
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$$Dom (f) = Im (f) = \mathbb{R}$$
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$$Dom (f) = \mathbb{R}$$, $$Im (f) =-1$$
- $$Dom (f) = \mathbb{R}$$, $$Im (f) = (-\infty, 3]$$, $$v=(2,3)$$