MHB Piecewise-defined Function....2

  • Thread starter Thread starter mathdad
  • Start date Start date
AI Thread Summary
To graph the piecewise-defined function y(x) = |x| for x ≤ 0 and y(x) = x^3 for x > 0, first simplify it to y(x) = -x for x ≤ 0 and y(x) = x^3 for x > 0. Plot the line y = -x on the interval (-∞, 0] and the cubic function y = x^3 on the interval (0, ∞). The graph is created by drawing each piece separately on the same xy-plane. Using graphing tools like Wolfram can facilitate this process. Understanding how to graph piecewise functions accurately is essential for visualizing their behavior.
mathdad
Messages
1,280
Reaction score
0
What is the easiest way to graph a piecewise-defined function by hand?

y = | x | if x is < or = 0...this is the upper piece

y = x^3 if x > 0...this is the bottom piece
 
Mathematics news on Phys.org
RTCNTC said:
What is the easiest way to graph a piecewise-defined function by hand?

y = | x | if x is < or = 0...this is the upper piece

y = x^3 if x > 0...this is the bottom piece

We are given:

$$y(x)=\begin{cases}|x|, & x\le0 \\[3pt] x^3, & 0<x \\ \end{cases}$$

Now, since we have by definition:

$$|x|=\begin{cases}-x, & x<0 \\[3pt] x, & 0\le x \\ \end{cases}$$

And:

$$0=-0$$

We may simplify the given function by writing:

$$y(x)=\begin{cases}-x, & x\le0 \\[3pt] x^3, & 0<x \\ \end{cases}$$

And so, to plot this function by hand, we would draw the line $y=-x$ on the interval $(-\infty,0]$ and the cubic $y=x^3$ on the interval $(0,\infty)$.

View attachment 6653
 

Attachments

  • piecewise_001.png
    piecewise_001.png
    2.8 KB · Views: 101
Are you saying we graph one piece at a time on the same xy-plane?
 
RTCNTC said:
Are you saying we graph one piece at a time on the same xy-plane?

Yes. :D
 
I use wolfram for all my graphs.
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top