Topic
Given the function \(h\left( x \right) = \left\{ \begin{array}{l}1\quad \quad x < 0\\2\quad \quad x \ge 0\end{array} \right.\)
a) Which of the following points is on the graph of the function above: \(A\left( {0;0} \right),B\left( { – 1;1} \right),C\left( { 2021;1} \right);D\left( {2022;2} \right)\)
b) Indicate two points on the graph of the above function with coordinates equal to 2
c) Indicate the point on the graph of the above function whose coordinate is -2022
Solution method – See details
Let\(f\left( x \right) = a{x^2} + bx + c\)
At \(x = {x_0}\) then \(f\left( {{x_0}} \right) = a{x_0}^2 + b{x_0} + c\)
Detailed explanation
a) + At \(x = 0 \Rightarrow y = h(x) = 2 \Rightarrow A\left( {0;0} \right) \notin h\left( x \right)\)
+ At \(x = – 1 < 0 \Rightarrow y = h(x) = 1 \Rightarrow B\left( { – 1;1} \right) \in h\left( x \right)\)
+ At \(x = 2021 > 0 \Rightarrow y = h(x) = 2 \Rightarrow C\left( {2021;1} \right) \notin h\left( x \right)\)
+ At \(x = 2022 > 0 \Rightarrow y = h(x) = 2 \Rightarrow D\left( {2022;2} \right) \in h\left( x \right)\)
b) We have \(h(x) = 2\) if \(x \ge 0\)
Therefore, all points with non-negative coordinates have a coordinate of 2.
The set of points with coordinate 2 is \(S = \left\{ {(a;2)|a \ge 0} \right\}\)
For example: \(E\left( {3;2} \right),G\left( {100;2} \right)\)
c) For \(x = – 2022 < 0\) then \(y = h(x) = 1 \Rightarrow H\left( { – 2022;1} \right)\)
