# Domain and Range A level Maths

Currently revising for upcoming exams and have found I am unable to do/understand how to calculate the domain and range of a function.

I have watched and read many articles on it but still don't understand how it works and how to calculate it.

Original post by max33456789
Currently revising for upcoming exams and have found I am unable to do/understand how to calculate the domain and range of a function.

I have watched and read many articles on it but still don't understand how it works and how to calculate it.

Domain is the set of values on which the function is defined.

There are restrictive functions which do not allow every number for inputs, e.g. $\sqrt{x}$ asks for the input to be non-negative, or $\dfrac{1}{x}$ asks for the input to be non-zero, or $\ln x$ asks for input to be positive. These facts contribute to finding the domain.

E.g. the function $\ln (1 - x^2)$ exists only if $1 - x^2 > 0$ which when solved gives $-1 < x < 1$ so this is the domain.
Original post by max33456789
Currently revising for upcoming exams and have found I am unable to do/understand how to calculate the domain and range of a function.

I have watched and read many articles on it but still don't understand how it works and how to calculate it.

What don't you specifically understand? Domain shows you what x values are valid for a 1 to 1 y range.
To find the domain of a function, you find the inverse of the function.
The notations are a pain, but they tend to describe something straightforward.

Note, if you are ever asked to sketch the domain or range of trig functions, it's probably going to be a lot easier if you just remember the shapes instead of trying to figure it out yourself.
Original post by MindMax2000
What don't you specifically understand? Domain shows you what x values are valid for a 1 to 1 y range.
To find the domain of a function, you find the inverse of the function.
The notations are a pain, but they tend to describe something straightforward.

Note, if you are ever asked to sketch the domain or range of trig functions, it's probably going to be a lot easier if you just remember the shapes instead of trying to figure it out yourself.

I'm not to sure really, like I understand what the concept is, domain is the available values to input and range is what you get out but whenever presented with a more challenging function such as f(x) = 2arccos -1, I have no idea where to start with calculating domain or range.

Original post by max33456789
I'm not to sure really, like I understand what the concept is, domain is the available values to input and range is what you get out but whenever presented with a more challenging function such as f(x) = 2arccos -1, I have no idea where to start with calculating domain or range.

I can give you some pointers/guidance if you can give me an example of a question that you're stuck on should you wish.
Original post by max33456789
I'm not to sure really, like I understand what the concept is, domain is the available values to input and range is what you get out but whenever presented with a more challenging function such as f(x) = 2arccos -1, I have no idea where to start with calculating domain or range.

Two ideas:
(i) Know the standard properties of functions. For instance, the domain of arccos is defined to be [-1,1], and range is defined to be [0,pi].
No cheats here - gotta know them by heart.
(ii) Learn curve sketching. If you know what the curve of arccos looks like, sketching it to get an idea what the domain/range of your f(x) (via transformations) should be is very useful.
I think you can avoid the common error of just plugging in the end points of the domain to get the range by simply sketching the curve first. For instance, the range of f(x)=x^2 over the domain [-1,2] is NOT [1,4].

By per MindMax's advice, specific problems would be helpful for TSR sake (though I'd say your arccos one is probably specific enough).
Original post by max33456789
I'm not to sure really, like I understand what the concept is, domain is the available values to input and range is what you get out but whenever presented with a more challenging function such as f(x) = 2arccos -1, I have no idea where to start with calculating domain or range.

For
2arccos(x)-1
do you know/understand what the domain and range of arccos(x) is? If so, the *2 and -1 should do a simple transformation of the range of arccos(x) and the domain would be unchanged. If necessary, just imagine running some "typical" numbers through the function / sketch it as tony suggests.
Original post by tonyiptony
Two ideas:
(i) Know the standard properties of functions. For instance, the domain of arccos is defined to be [-1,1], and range is defined to be [0,pi].
No cheats here - gotta know them by heart.
(ii) Learn curve sketching. If you know what the curve of arccos looks like, sketching it to get an idea what the domain/range of your f(x) (via transformations) should be is very useful.
I think you can avoid the common error of just plugging in the end points of the domain to get the range by simply sketching the curve first. For instance, the range of f(x)=x^2 over the domain [-1,2] is NOT [1,4].

By per MindMax's advice, specific problems would be helpful for TSR sake (though I'd say your arccos one is probably specific enough).

Thank you for explaining this, my teacher didn't really teach us this topic and we are expected to know it so trying to wrap my head around it.

As for learning the domain for arccos as (-1,1) what is the domain for arctan and arcsin? Are they the same?

Definitely seems alot clearer now thanks
Original post by max33456789
Thank you for explaining this, my teacher didn't really teach us this topic and we are expected to know it so trying to wrap my head around it.

As for learning the domain for arccos as (-1,1) what is the domain for arctan and arcsin? Are they the same?

Definitely seems alot clearer now thanks

You should know about inverse functions sqrt(), arcsin(), arccos(), ...

sqrt(x) has a domain which is x>=0 and the range is >=0, so it returns a positive value. Its not defined for negative arguments. The original function x^2 is increasing on the domain x>=0 and the "problem" is if the domain include the turning point at 0 and a portion of x<0 as then x^2 is not invertible as two inputs say +/-1 both map to 1. Hence the domain/range of sqrt() is the positive numbers.

arcsin(x) and arccos(x) are similar. Theyre only defined for an increasing or decreasing segment of the original functions sin() and cos() otherwise the inverse does not exist if the domain includes a turning point. So arcsin(x) has a range -pi/2 to pi/2 and arcos(x) has a range 0 to pi. Both can be easily seen if you sketch the original sin and cos curves and think about which section of the curve is increeasing/decreasing, includes 0 and covers the range -1 to 1. It underpins the cast diagram, the multiple solutions, ... so you should "know" this.
(edited 8 months ago)