# Piecewise Function: Intervals of Increase/Decrease & Extremes/Asymptotes

• Kolika28
In summary: No, the oblique tangent is a line on the form kx+m, not a function. It cannot be written as x2+1x2+1\frac{x}{2}+1. You need to use the method of limits to find the slope of the oblique tangent.
Kolika28
Homework Statement
##f(x)=\left\{
\begin{array}{ll}
\frac{e^{-x}-1}{x}, & x>0 \\
\frac{x}{2}+1, & x\leq 0 \\
\end{array}
\right.##

a) At which intervals are f strictly increasing and at what intervals are f strictly decreasing.
b) Determine any local and global extreme values for f.
c) Determines if f has asymptotes.
Relevant Equations
The derivative
a) At which intervals are f strictly increasing and at what intervals are f strictly decreasing.

Should I just find the derivative of both of the functions? If so, I get that the function is increasing at the intervals (−∞,0) and (0,∞). Is this right, or can I just say that the function is increasing at the interval (−∞,∞)?

b) Determine any local and global extreme values for f.

When graphing the function I don't see any local or global extreme values. f(x) consists of a straight line and curve with where f(x)=0 is not true. The whole function is not bounded, so I can't look at the values in the endpoints. But my teacher says there are extreme values. But how so?

c) Determines if f has asymptotes.

I know there is at least one horisontal asymptote, y=0, given the first function. Because

I was told there is one oblique asymptote also. But how?

Delta2
a) You would just say that the function is strictly increasing everywhere.

b) The function is bounded from above, but it has no extreme points. (It has a supremum, but no maximal value.)

c) what happens for large negative x?

Orodruin said:
a) You would just say that the function is strictly increasing everywhere.

b) The function is bounded from above, but it has no extreme points. (It has a supremum, but no maximal value.)

c) what happens for large negative x?

b) Should I just say that it has a supremum then in x=0??
c) For large negative x I look at ##\frac{x}{2}+1##. And I just get ##-∞## . But that does not tell me what the oblique slope is?

Orodruin said:
a) You would just say that the function is strictly increasing everywhere.

Not on the interval [-1,1].

b) The function is bounded from above, but it has no extreme points. (It has a supremum, but no maximal value.)

##f(0) = 1## is an absolute maximum.

Last edited:
Delta2
LCKurtz said:
Not on the interval [-1,1].
##f(0) = 1## is an absolute maximum.
Oops. I just assumed OP had checked for continuity at x=0...

Kolika28 said:
b) Should I just say that it has a supremum then in x=0??
No, it has a global max in x=0.

Kolika28 said:
c) For large negative x I look at x2+1x2+1\frac{x}{2}+1. And I just get −∞−∞-∞ . But that does not tell me what the oblique slope is?
You are not looking for a value of the function. You are looking for a line on the form ##kx+m## that the function approaches.

Kolika28 said:
b) Should I just say that it has a supremum then in x=0??
c) For large negative x I look at ##\frac{x}{2}+1##. And I just get ##-∞## . But that does not tell me what the oblique slope is?
The slope of a line is constant, here =1/2.

Ok, now I understand what you mean by task b. But I'm still confussed about c). So I'm supposed to find a line on the form ##kx+m##.
Orodruin said:
You are not looking for a value of the function. You are looking for a line on the form kx+mkx+mkx+m that the function approaches.
WWGD said:
The slope of a line is constant, here =1/2.
So is the oblique tanget just ##\frac{x}{2}+1##?

LCKurtz said:
Not on the interval [-1,1].
Hmm, why not this interval. Is the function decreasing here?

Kolika28 said:
Hmm, why not this interval. Is the function decreasing here?
Not in the entire interval, no. You have (correctly) concluded that the function is increasing on (-infinity,0) and (0,infty). What is left to check?

Orodruin said:
Not in the entire interval, no. You have (correctly) concluded that the function is increasing on (-infinity,0) and (0,infty). What is left to check?
I'm sorry, but I'm really blank right now. I don't see what's left to check. Sorry, for all the questions by the way, I just really want to understand!

What is the value of f(0)? What is the value of f(0.000000001)?

Kolika28
Orodruin said:
What is the value of f(0)? What is the value of f(0.000000001)?
Ohh, I understand now. Thank you so much!My last question is:
Kolika28 said:
So is the oblique tanget just x2+1x2+1\frac{x}{2}+1?

## 1. What is a piecewise function?

A piecewise function is a function that is defined by different rules or formulas over different intervals. This allows for the function to have different behaviors in different parts of its domain.

## 2. What are intervals of increase and decrease?

Intervals of increase and decrease are the intervals in which a function is either increasing or decreasing. In other words, the function's output values are either getting larger or smaller as the input values increase.

## 3. What are extremes of a piecewise function?

The extremes of a piecewise function are the highest and lowest points of the function, also known as the maximum and minimum values. These points can occur at the endpoints of each interval or at critical points within the intervals.

## 4. What are asymptotes in a piecewise function?

Asymptotes are lines that a function approaches but never touches. In a piecewise function, there can be asymptotes in different intervals, depending on the behavior of the function in that interval. Asymptotes can be horizontal, vertical, or oblique.

## 5. How do I find the intervals of increase and decrease, as well as the extremes and asymptotes, of a piecewise function?

To find the intervals of increase and decrease, you can graph the function and observe where the function is increasing or decreasing. To find the extremes, you can take the derivative of the function and find the critical points, then check the value of the function at those points. Asymptotes can be found by analyzing the behavior of the function at the endpoints of each interval and at points of discontinuity.

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