Finding Domain of f(x) & g(x) Equations

  • Thread starter xnonamex0206
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For example, if f(x)= x and g(x)= 1/x, both f and g are defined everywhere except x= 0. fg(x)= x(1/x)= 1 is defined for all x except x= 0. f+g(x)= x+ 1/x is defined everywhere except x= 0. f-g(x)= x- 1/x is defined everywhere except x= 0. f/g(x)= x/(1/x)= x^2 is defined everywhere except x= 0.In summary, the functions (f+g)(x), (f-g)(x), (fg)(x), and (f/g)(x) can all be simplified to real
  • #1
xnonamex0206
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Homework Statement



Find a. (f+g)(x) b. (f-g)(x) c. (fg)(x) d. (f/g)(x)

f(x)=2x-5 & g(x)=4

Homework Equations



(f+g)(x)=2x-1
(f-g)(x)=2x-9
(fg)(x)=8x-20
(f/g)(x)=2x-5/4

The Attempt at a Solution


When I've completed all the other equations, I'm stuck on domain. I'm thinking that its all reals, but I'm not quite sure on how this is true. Is it because it cannot equal zero, or a non-negative number?

In other other words how exactly do you find the domain?
 
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  • #2
Just a little note, they are actually asking for the *largest* possible domain of each function over the reals, as you can give a function any domain you want, as long as the function is defined for every element of its domain.
Is there any value of x for which any of your functions does not yield a real number? If they exist, these x's cannot be in any domain of your function.
Hint: The identity function I(x) = x is defined for all real numbers x. Your functions are just real numbers added to real multiples of this function. Ie., it boils down to showing that if x is a real number, then kx is also a real number for real k.
PS. Your (f/g)(x) is missing the division by 4.
 
  • #3
fg, f+g and f-g are defined everywhere, where f and g are both defined. f/g is defined everywhere, where f and g are both defined and g does not equal zero.
 
  • #4
slider142 said:
Just a little note, they are actually asking for the *largest* possible domain of each function over the reals, as you can give a function any domain you want, as long as the function is defined for every element of its domain.
Is there any value of x for which any of your functions does not yield a real number? If they exist, these x's cannot be in any domain of your function.
Hint: The identity function I(x) = x is defined for all real numbers x. Your functions are just real numbers added to real multiples of this function. Ie., it boils down to showing that if x is a real number, then kx is also a real number for real k.
PS. Your (f/g)(x) is missing the division by 4.

What does the k represent? So if x is a real number then the domain should include all reals? So basically to find the domain you should plug in some x's to see if they aren't real numbers?

I fixed the (f/g) forgot to add the /4.

fg, f+g and f-g are defined everywhere, where f and g are both defined. f/g is defined everywhere, where f and g are both defined and g does not equal zero.

What!?
 
  • #5
Yes, xnoname0206 is correct. If the domains of f and g are not the same, the domains of fg, f+ g and f-g are the intersections of the domains of f and g. That is what xnoname0206 meant by "where f and g are both defined". f/g is defined on the intersection of the domains minus points where g(x)= 0.
 

1. What is the definition of a domain in a function?

The domain of a function is the set of all possible input values, or the independent variable, for which the function is defined. In other words, it is the set of all values that can be plugged into the function to produce a valid output.

2. How do you find the domain of a function algebraically?

To find the domain of a function algebraically, you must identify any values that would make the function undefined. This includes any values that would result in division by zero, negative numbers under a square root, or numbers that would make the function undefined in some other way. Once you have identified these values, you can write the domain using interval notation.

3. What are some common types of functions and their domains?

Linear functions have a domain of all real numbers. Quadratic functions also have a domain of all real numbers, unless there is a restriction on the variable. Rational functions have a domain of all real numbers except for any values that would result in division by zero. Exponential and logarithmic functions have a domain of all real numbers.

4. How do you find the domain of composite functions?

To find the domain of a composite function, you must first find the domain of each individual function within the composite function. Then, you must determine the values that would make the entire composite function undefined. These values would be excluded from the domain of the composite function.

5. Can the domain of a function be infinite?

Yes, the domain of a function can be infinite. This is often the case with exponential and logarithmic functions, which have a domain of all real numbers. Additionally, some functions may have an infinite domain if they have a repeating pattern or if the function is defined for all real numbers.

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