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This is one of our many quizzes on piecewise functions. Click on this to access more such competitive quizzes on functions.

Piecewise functions Quiz 2

mathematics

 Consider the following equations carefully and find out that which of the following is not a piecewise function?

  Options (b) and (c) will form piecewise functions. By tracing the graph of equation in option b we find that a staircase graph will be formed. This graph will have gaps in between its traces. Similarly, the equation in option (c) will form a graph which will not be continuous at point $x = 0$. But the equation in option (a) $\\ y=2x-\left| x \right|,\,\,\forall x\in \mathbb{R}$ is a continuous function because of its continuity in the graph. $\\  x=0\Rightarrow y=2\left( 0 \right)-\left| 0 \right|=0\Rightarrow \left( 0,0 \right) \\   x=1\Rightarrow y=2\left( 1 \right)-\left| 1 \right|=2-1=1\Rightarrow \left( 1,1 \right) \\$ Similarly, by substituting the points in the equation we get the following graph. As there are no gaps in the following graph so, the graph is continuous.$ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/bfa2a406-3999-4ac2-8d57-b8e553e8fde1935630802440943329.png"/>

 Figure out the correct equation of the following graph.<img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/3ffe8139-a204-4518-a644-9bd7522458f83834602139355090970.png"/>

 Consider the equations one by one. Substitute the values of $x$ and trace the graph accordingly. The one which coincides with the given graph results in the correct option.

  Since, the options carry trigonometric functions try to substitute $x = 0$ in all the equations. $ \\   \Rightarrow y=\sin \left( 0 \right)+\cos \left( 0 \right)=1\left( \text{Since,}\sin \left( 0 \right)=0,\cos \left( 0 \right)=1 \right) \\$ Similarly, by substituting points in $x$ and cross checking them with the given graph, we get that the graph of function. $\sin \left( x \right)+\cos \left( x \right)\,\,;\,\,\forall x\in \left[ -0.7,3 \right]$ seems to be coinciding with the green colored trace in the given graph. Also, after following the process of substitution in $x$, the graph of function ${{x}^{2}}-7x-0.1y=-12\,\,;\,\,\forall x\in \left[ 3,3.5 \right]$  coincides with the black trace in the given graph.

 Try to figure out the equation of the following graph.<img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/13cab915-0a77-45db-a33f-7c75acae626a4202636693163325555.png"/>

 Consider equations one by one. Draw the traces of each equation by taking the given values of $x$ simultaneously. If the graph is coinciding with the graph in the question, then it will be the right answer.

 <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/0d7e1e29-d20a-4fc9-b288-51c611e30e987780640134181250805.png"/> $\\$ It is clearly visible that the following graph must have three different equations. So, we will consider the options one by one. Substitute points of $x$ in their respected equations. Doing so, we get a trace after plotting them. After drawing the graph, only $y=\left\{ \left. \begin{align}  & -{{x}^{2}}+2\,\,\,\,\,\,\,\,\,\,\,;\forall x\in \left( \left. -2.9,-2.4 \right] \right. \\  & 2x+1\,\,\,\,\,\,\,\,\,\,\,;\forall x\in \left( \left. -2.4,1 \right) \right. \\  & {{x}^{2}}+2\,\,\,\,\,\,\,\,\,\,\,;\forall x\in \left[ \left. 2,4.7 \right) \right. \\ \end{align} \right\} \right.$ coincides with the given graph.

 Find the value of $\Omega $ if the functions are given by $f\left( x \right)=\sin \left( \log \left( x \right) \right)$ and $g\left( x \right)=\left\{ \begin{align}  & \dfrac{{{x}^{2}}+24}{x-23}\,\,\,\,\,\,\,\,\,\,\,;x\ne 23 \\  & \Omega \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,;x=23 \\ \end{align} \right\}$ where $g\left( x \right)$ is a piecewise function.

 Since, $g\left( x \right)$  is piecewise function, so we need to consider that point to which we can equate the function $f\left( x \right)$. Consider point $x = 23$ then, to find the value of $\Omega $ equate $g\left( 23 \right)=\Omega $ to $f\left( 23 \right)$.

 $\text{The equation }g\left( x \right)\text{ is not continuous at x=23. So, we equate }g\left( 23 \right)\text{ with the function }f\left( 23 \right). \\   \therefore f\left( 23 \right)=g\left( 23 \right) \\  \Rightarrow \sin \left( \log \left( 23 \right) \right)=\Omega  \\   \Rightarrow 0.978=\Omega  \\$ 

 Which of the following points satisfies the equation $y=\left[ x+2 \right]$?

 Use greatest integer function techniques to solve in which the greatest integer function $\left[ x \right]$ is always the number that is a round-off of the number $x$ to an integer with a condition that the resulting round-off number must be an integer and less than $x$.

 $y=\left[ x+2 \right] \\  \Rightarrow y=\left[ 4.3 \right] \\$ The greatest integer function $\left[ x \right]$ is always the number that is a round-off of the number $x$ to an integer with a condition that the resulting round-off number must be an integer and less than $x$. $\\ 4<4.3\,\text{and}\,4\in \mathbb{Z} \\   \Rightarrow y=4 \\$ 