# Piecewise function

## 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

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Orodruin
Staff Emeritus
Homework Helper
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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?

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?

LCKurtz
Homework Helper
Gold Member
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
Orodruin
Staff Emeritus
Homework Helper
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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...

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

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.

WWGD
Gold Member
2019 Award
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##.
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.
The slope of a line is constant, here =1/2.
So is the oblique tanget just ##\frac{x}{2}+1##?

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

Orodruin
Staff Emeritus
Homework Helper
Gold Member
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?

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!

Orodruin
Staff Emeritus
Homework Helper
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What is the value of f(0)? What is the value of f(0.000000001)?

• Kolika28
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:
So is the oblique tanget just x2+1x2+1\frac{x}{2}+1?