The image contains a piecewise function, $ f(t) $, which is defined as follows:- For $ 0 \leq t < 1 $, $ f(t) = e^{-(t-1)} $.- For $ t \geq 1 $, $ f(t) = t^3 $.The task is to express this function as a step function.

A step function can be written in the form:for all real numbers x.If n ≥ 0, αi are real numbers and…

Answered: Write this function as a step function…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

The image contains a piecewise function, $ f(t) $, which is defined as follows:

- For $ 0 \leq t < 1 $, $ f(t) = e^{-(t-1)} $.
- For $ t \geq 1 $, $ f(t) = t^3 $.

The task is to express this function as a step function.

Expert Solution

Step 1: Step Function Definition

A step function $f colon straight real numbers rightwards arrow straight real numbers$ can be written in the form:

$f open parentheses x close parentheses equals sum from i equals 0 to n of alpha subscript i chi subscript A subscript i end subscript left parenthesis x right parenthesis$

for all real numbers x.

If n ≥ 0, α_i are real numbers and A_i are intervals, then the indicator function of A is χ_A, and it can be written as below:

$chi subscript A left parenthesis x right parenthesis open curly brackets table row cell 1 semicolon end cell cell i f space x element of A end cell row cell 0 semicolon end cell cell i f space x not an element of A end cell end table close$