how to tell if a function is defined?

Hi guys

how do you tell if a function is defined for all values of x and which y functions are defined

eg y = e^-x

Depends on the function, you need to make sure you're not taking the square root of a complex number (for a real valued function), you need to make sure you're not dividing by 0, that you're not taking the logarithm of a non-positive number, etc...
(This is strictly A-Level stuff, generally a function is defined with its domain given already)

So, for example: If you had

\ln (x+1)

then you know that you always need

x+1 > 0 \implies x > -1

, if you had

\frac{1}{x+3}

then you know you need

x+3 \neq 0 \implies x \neq -3

.

If you had

\sqrt{5x+2}

then you know that

5x + 2\geq 0 \implies x \geq -\frac{2}{5}

, etc...

For the example you've given, it's defined for all real x because no problems would be caused by any value of x, there wouldn't be any square rooting of negative number, logarithms of negative numbers, divisions by 0, etc...

Reply 2

TeeEm

Original post by madmadmax321

Hi guys

how do you tell if a function is defined for all values of x and which y functions are defined

eg y = e^-x

Depends on the structure of the function

you may not divide by zero
arguments of radicals must be non negative
arguments of logarithms must be positive
base of logarithms must be positive and must not be one
bases of exponentials must be non negative
arguments of arcsin, arccos must be between -1 and 1 inclusive
etc

Reply 3

username853993

Original post by Zacken

\ln (x+1)

then you know that you always need

x+1 > 0 \implies x > -1

, if you had

\frac{1}{x+3}

then you know you need

x+3 \neq 0 \implies x \neq -3

.

If you had

\sqrt{5x+2}

then you know that

5x + 2\geq 0 \implies x \geq -\frac{2}{5}

Thank you for the reply!

I think I understand

so for (x^2)/(x^2+1) x would always be defined but for 1/-sqrt(1-x^2) they would not

Reply 4

username853993

Original post by Zacken

\ln (x+1)

then you know that you always need

x+1 > 0 \implies x > -1

, if you had

\frac{1}{x+3}

then you know you need

x+3 \neq 0 \implies x \neq -3

.

If you had

\sqrt{5x+2}

then you know that

5x + 2\geq 0 \implies x \geq -\frac{2}{5}

Also when the question says 'for which y are the functions defined (real values of y)?

does that mean are they possible in positive or negative y?

Reply 5

Mpagtches

Look in the dictionary :p:

Reply 6

Zacken

Original post by madmadmax321

Thank you for the reply!

I think I understand

so for (x^2)/(x^2+1) x would always be defined but for 1/-sqrt(1-x^2) they would not

A function can be defined for a certain value of x and not defined for other values.

The first example you gave will be defined for every value of x.

The second example will defined everywhere except where

x \geq 1

, at that point, the function is not defined because you are dividing by 0 for x=1 and and you are square rooting a negative number for x > 1, both now allowed. So you can say that the function is defined for

x < 1

Reply 7

username853993

Original post by Zacken

x \geq 1

x < 1

Sorry I meant it wouldn't be defined for all values of x

Reply 8

Zacken

Original post by madmadmax321

Sorry I meant it wouldn't be defined for all values of x

Saying that is not enough, you should state for what values of x it is defined for.

Otherwise, it's like solving an equation and writing the answer as "it's not all the numbers, just a few." - you need to give the exact range of values for which the function is defined, just like you give the exact numbers that are solutions to an equation.

Reply 9

username853993

Original post by Zacken

so I need to state it even if the question is just asking 'is the function defined for all values of x?'

note I am not doing A-level maths, it was a part of my foundation year lectures that I missed

Reply 10

Zacken

Original post by madmadmax321

so I need to state it even if the question is just asking 'is the function defined for all values of x?'

note I am not doing A-level maths, it was a part of my foundation year lectures that I missed