1. ## Specify the Domain

Specify the domain of each variable.

1. ax + b

Here, x is the variable. I say a and b are constants.
Therefore, the domain is ALL REAL NUMBERS.
Is this right?

2. ax^(1/3) + b

Again, x is the variable. I know that a and b are constants.
I decided to rewrite question 2 as a*cuberoot (x) + b. Is this correct? What would be the domain in this case?

2. ## Re: Specify the Domain

1. ax + b is linear , and hence domain is all real x and range all real y

2. $x^{\frac13}=\sqrt[3]{x}$, so yes, that is a correct rewrite

if you consider the basic cube root curve $y=\sqrt[3]{x}$ its domain is all real x and its range is all real y.
by multiplying it by a, you are essentially only affecting the 'narrowness' or 'width' of the curve, and adding b will move it up or down.
neither of these operations will have any impact on the domain or range of the cubic

3. ## Re: Specify the Domain

Originally Posted by jacs
1. ax + b is linear , and hence domain is all real x and range all real y

2. $x^{\frac13}=\sqrt[3]{x}$, so yes, that is a correct rewrite

if you consider the basic cube root curve $y=\sqrt[3]{x}$ its domain is all real x and its range is all real y.
by multiplying it by a, you are essentially only affecting the 'narrowness' or 'width' of the curve, and adding b will move it up or down.
neither of these operations will have any impact on the domain or range of the cubic
Can you graph y = cuberoot (x) and show me how to determine the domain and range from the graph itself?

4. ## Re: Specify the Domain

the blue curve is $y=x^3$

the red curve is $y=\sqrt[3]x$

you can see the the cube root curve is very similar in shape to the cubic and are actually inverse functions

5. ## Re: Specify the Domain

To determine the domain for a function, f(x), look for values where f cannot be applied. Given f(x)= ax^(1/3)+ b, we can take the cube root of any number, we can multiply any number by a and we can add b. The domain is 'all numbers'. To determine the range, solve the equation for x. From y= ax^(1/3)+ b, ax^(1/3)= y- b, x^(1/3)= y/a+ b/a, and, finally, x= (y/a+ b/a)^3. For any value of y we can: divide by a, add b/a, and then cube. The range is "all numbers'.

A more interesting function is y= x^2. we can, of course, square any number so the domain is 'all numbers'. But solving for x involves taking the square root of y. We can only take the square root (x and y are real numbers here) of non-negative numbers. The range is 'all non-negative numbers".

6. ## Re: Specify the Domain

Originally Posted by jacs

the blue curve is $y=x^3$

the red curve is $y=\sqrt[3]x$

you can see the the cube root curve is very similar in shape to the cubic and are actually inverse functions

7. ## Re: Specify the Domain

Originally Posted by HallsofIvy
To determine the domain for a function, f(x), look for values where f cannot be applied. Given f(x)= ax^(1/3)+ b, we can take the cube root of any number, we can multiply any number by a and we can add b. The domain is 'all numbers'. To determine the range, solve the equation for x. From y= ax^(1/3)+ b, ax^(1/3)= y- b, x^(1/3)= y/a+ b/a, and, finally, x= (y/a+ b/a)^3. For any value of y we can: divide by a, add b/a, and then cube. The range is "all numbers'.

A more interesting function is y= x^2. we can, of course, square any number so the domain is 'all numbers'. But solving for x involves taking the square root of y. We can only take the square root (x and y are real numbers here) of non-negative numbers. The range is 'all non-negative numbers".

8. ## Re: Specify the Domain

Looking at the red curve.... is there anything that x can't be? No, it is a continuous graph that extends infinitely from left to right. So there is no restriction on what x can be. Therefore the domain is "all real numbers".

Similarly for y. As you look at the curve from top to bottom, y can be any real number. (Note: the graph extends upwards to the right and downwards to the left indefinitely) Hence the range is "all real numbers".

9. ## Re: Specify the Domain

Originally Posted by Debsta
Looking at the red curve.... is there anything that x can't be? No, it is a continuous graph that extends infinitely from left to right. So there is no restriction on what x can be. Therefore the domain is "all real numbers".

Similarly for y. As you look at the curve from top to bottom, y can be any real number. (Note: the graph extends upwards to the right and downwards to the left indefinitely) Hence the range is "all real numbers".
Now I get it. All real numbers can be written as
(-infinity, infinity) in interval notation, right?